1. Manual

1.1. Overview

1.1.1. Methods

XMVB provides an ab initio computing platform for various VB approaches, including classical VB methods, such as VBSCF, BOVB, VBCI, VBPT2, modern VB methods, such as SCVB and GVB, and molecular orbitals based VB method, BLW. Combined with solvation models, it can perform VBPCM, VBEFP, and VBSMD to account for solvent effects. Incorporating XMVB with KS-DFT code, it can be applied to hybrid DFVB calculation. In this manual, only a brief introduction to classical VB methods is provided. Please read the following references for details:

Articles:

Book Chapters:

1.1.1.1. The VBSCF method

The wave function of Valence Bond Self Consistent Field (VBSCF) method is the linear combination of VB structures, as shown below.

\[\Psi = \sum_{K}C_K\Phi_K\]

In VBSCF method, All VB structures share the same set of VB orbitals, and both sets of the structure coefficients and VB orbitals are optimized simultaneously to minimize the total energy. This is comparable to the MCSCF method in the MO theory. VBSCF method takes care of the static electron correlation and gives equivalent results to the MO-based CASSCF calculations. It should be noted that the dynamic electron correlation is not accounted for in the VBSCF method. In XMVB, VBSCF method is the default method, thus this keyword can be ignored.

1.1.1.2. VB Methods including Dynamic Correlation

The VBSCF result includes only static correlation energy, which makes VBSCF results not accurate enough for quantitative researches. The purpose of post-VBSCF methods is to take dynamic correlation into account as much as possible to get accurate enough results. There are several post-VBSCF methods developed so far and will be introduced in this section. It is strongly recommended to perform post-VBSCF calculations with initial guesses from a pre-proceeded VBSCF calculation. As to VBCI and VBPT2, this is enforced.

1.1.1.2.1. The BOVB method

The orbitals of Breathing Orbital Valence Bond (BOVB) method are also optimized by SCF procedure, as VBSCF does. The difference between VBSCF and BOVB methods is that BOVB provides an extra degree of freedom during orbital optimization. In BOVB method, each VB structure has its own set of orbitals and are optimized independently

\[\Psi^{\textrm{VBSCF}} = C_1\left( \vert \phi_a\overline{\phi_b} \vert - \vert \phi_b\overline{\phi_a} \vert \right) + C_2\vert \phi_a\overline{\phi_a} \vert + C_3 \vert \phi_b\overline{\phi_b} \vert\]

\[\Psi^{\textrm{BOVB}} = B_1\left( \vert \phi_a\overline{\phi_b} \vert - \vert \phi_b\overline{\phi_a} \vert \right) + B_2\vert \phi'_a\overline{\phi'_a} \vert + C_3 \vert \phi'_b\overline{\phi'_b} \vert\]

Thus, the orbitals adopt themselves to the instantaneous field of the VB structures, rather than to the mean field of all the structures in VBSCF. This degree of freedom makes the orbitals in BOVB “breathing” in different structures, introduces dynamic correlation, and thereby improves considerably the accuracy of VB computations.

1.1.1.2.2. The VBCI method

The VBCI method is based on localized VB orbitals. In this method VB orbitals are divided to several blocks (occupied and virtual orbitals). Excited VB structures are generated by replacing occupied VB orbitals with virtual orbitals that are localized on the same block. The wave function of VBCI is the linear combination of all reference and excited VB structures

\[\Psi^{\textrm{VBCI}} = \sum_K\sum_iC_{Ki}\Phi^i_K\]

where $\Phi^i_K$is CI structure coming from VBSCF structure K, including reference and excited structures, and the coefficients ${C_{Ki}}$ are obtained by solving the secular equation. The VBCI weight can be given either with equation

\[W_K = C_K\sum_LC_LM_{KL}\]

which gives weights of all CI structures, or in a more compact way as

\[W_K = \sum_iW_{Ki}\]

where $W_K$ is the contracted weights of reference structure K, including weights of all CI structures coming from structure K.

Allowing for different excitations for different electronic shells, currently the VBCI method consists of the following calculation levels:

VBCI(S,S): only single excitations are involved in either active electron or inactive electron. In brief, this is a VBCIS procedure.
VBCI(D,S): the active shell is treated by single and double excitations, whereas the inactive shell by single excitations only. Also included in this level are double excitations which consist of a single excitation from each shell.
VBCI(D,D): single and double excitations are involved for both active and inactive electrons, in short, VBCISD.

1.1.1.2.3. The VBPT2 Method

Another post-VBSCF method is Valence Bond second-order Perturbation Theory (VBPT2) method. The wave function of VBPT2 can be separated into 2 parts as

\[\Psi^{\textrm{VBPT2}} = \Psi^0 + \Psi^1\]

where VBSCF wave function is taken as the zeroth-order wave function $\Psi^0$, and the first-order part is the combination of singly and doubly excited wave functions

\[\Psi^1 = \sum_{R\in V^{SD}}C^1_R\Phi_R\]

To enhance the efficiency of VBPT2, the virtual orbitals are delocalized and orthogonal to the occupied space, and the excitations include all virtual orbitals. In this manner, the excited structures in VBPT2 don’t belong to any fundamental structure, and the matrix elements can be calculated easily with Coulson-Slater rules.

1.1.2. Installation

Both distributions of XMVB are currently available for LINUX platform. 1.5GB RAM is required. Followings are the instructions for installation. Note that the source code will NEVER be released to the users. Only compiled object file or executable files are available for users.

1.1.2.1. Module distribution

Tip

This is an installation guidance for GAMESS-XMVB,whose functions are not provided on the XACS cloud computing platform.

To build the module distribution, the user may need a library file libgamess-xmvb.a which can be found in the package they obtained. The user may also need to prepare LIBCINT library file libcint.a which is needed for some functions in XMVB. After that, copy these two library files to object directory in GAMESS-US. The user also need to make the following modifications to linke XMVB into GAMESS-US package. We assume that the user will run GAMESS-XMVB with AMD-64 Linux, which corresponds to linux64 architecture of GAMESS-US, and the target is sockets. All operation is proceeded in the root directory of GAMESS-US.

Open ddi/compddi and modify

if($COMM == sockets) then
    set DDI_COMM = '-DDDI_SOC'
endif

to

if($COMM == sockets) then
    set DDI_COMM = '-DDDI_SOC -mcmodel=large'
endif

then run

cd ddi && ./compddi

to recompile DDI.

Open comp, find the entry for linux64, then modify
```
set EXTRAOPT=" "
```
to
```
set EXTRAOPT="-mcmodel=large"
```
in the branch of gfortran, then run
```
./compall
```
to recompile all modules.
Open lked, modify
```
set XMVB=false
```
to
```
set XMVB=true
```
to activate linking XMVB, then modify
```
set VBOBJ='xmvb.o'
```
to
```
set VBOBJ='libgamess-xmvb.a libcint.a /path/to/lapack /path/to/blas'
```
where /path/to/lapack is the location (absolute path) of LAPACK library file, and /path/to/blas is the location of BLAS library file. If MKL is used for GAMESS-US, LAPACK and BLAS library files may be omitted.
Open lked, find the entry for linux64, then modify
```
set LDOPTS=' '
```
to
```
set LDOPTS='-fopenmp -mcmodel=large'
```
in the branch of gfortran, then run
```
./lked gamess xmvb
```
to start linking. After that, an executable file gamess.xmvb.x including the whole capability of XMVB will be generated.
Before running GAMESS-XMVB, don’t forget to append a line
```
setenv  VBINP $USERSCR/$JOB.xmi
```
in the file gms-files.csh. The varialbe $USERSCR can be replaced with other values such as $SCR which is defined in rungms.

1.1.2.2. Stand-alone distribution

The stand-alone distribution is released as a package of compiled executable files. To install the stand-alone distribution, the users should:

Unpack the compressed tar file by using the following command,
tar xvfz xmvb.tar.gz
Once the file is unpacked successfully, a new directory xmvb/ will be created.
Set the VBDIR environment variable to the location of XMVB package and append xmvb/ to your $PATH. The variable VBDIR is essential for PPD algorithm and utility PREINT.

1.1.3. Running a job

A typical XMVB job consists of the following two steps:

Prepare the integrals of primitive basis functions and the MO orbitals of the system. An input file including molecular information and basis set information is needed. Files “1e.tem”, “2e.tem” and “orb.mo” will be generated after this step.
Run XMVB calculation. A file with extension name “.xmi”(XMI file, see section Input ) is needed here.

The programs, utilities and files needed in these 2 steps differ in different distributions. The details are shown below.

1.1.3.1. For module distribution

The two steps of an XMVB job in module distribution are done in one shot with command

rungms job [VERNO] [1]

after all files are prepared.

Step 1 is done in GAMESS program with a GAMESS input file including line

$CONTRL VBTYP=XMVB $END

XMVB calculation will be proceeded automatically following the completion of step 1.

Tip

By default, XMVB guess and input files will not be copied into the $SCR directory, which is the real directory where GAMESS- US is doing the calculation. Please copy these files into your $SCR directory BEFORE the calculation. Otherwise an error may occur.
After the calculation, the integral files and XMVB outputs will also be left in $SCR. Integral files may be used for XMVB calculations with stand-alone XMVB program. Please remember to get them if you need them later for stand-alone XMVB.

1.1.3.2. For stand-alone distribution

With stand-alone distribution used, step 1 is done with program PREINT. After obtaining the integral, create an input file file.xmi for your job (for details, see section Input). Then run the XMVB job by typing command

xmvb file.xmi

Then an output file with name file.xmo will be obtained. For use XMVB in parallelization, see below.

1.1.3.3. Parallelization in stand-alone XMVB

MPI and OpenMP are two popular protocols for parallelization. MPI will create a bunch of processes and the data is synchronized by message-passing, and is useful for multi-node parallelization in distributed systems, e.g. clusters. OpenMP creates a bunch of lightweight threads under a process and is useful for parallelization in single node. The MPI+OpenMP strategy may take the advantages of both protocols to provide a highly-efficient parallelization with multiple nodes. With $N_\textrm{MPI}$ MPI processes and $N_\textrm{OMP}$ OpenMP threads for each process, the total number of CPU cores consumed for a parallel job $N_\textrm{CPU}$ is

\[N_\textrm{CPU} = N_\textrm{MPI} * N_\textrm{OMP}\]

Currently such strategy is supported only by stand-alone XMVB. To use MPI in XMVB, the user may have MPICH installed in the computer. The version of MPICH to compile XMVB is 3.3.2.

1.1.3.3.1. MPI parallelization

To run XMVB with MPI parallelization, the user need to type the command

mpirun -np N xmvb file.xmi

to start a computation with N processes. For more information about running parallel computation with MPICH, please refer to the MPICH documentation.

1.1.3.3.2. OpenMP parallelization

The OpenMP parallelization does not need to change the command of running XMVB jobs, as shown in 3.2. The number of CPU cores for the parallelization can be controlled by setting the variable OMP_NUM_THREADS

export OMP NUM THREADS=N

if you just need a certain number N of CPUs for your job. If the variable is abesent, OpenMP parallelizes the XMVB computation with ALL processors in your computer.

For large systems, OpenMP parallelzation may proceed a strange “segmentation fault”. This is because the stack size of threads is not large enough. This can be avoided by setting the stack size to a certain number to avoid this error. In OpenMP parallelization, the stack size of master and slave threads are set in different ways. The stack size of master thread is set by command ulimit as shown

ulimit -s stack_size

The default stack size is 8192. Setting a larger value or simply

ulimit -s unlimited

The stack size of slave threads are controlled by environment variable $OMP_STACKSIZE. Following command will set the stack size of each slave threads to 1GB

export OMP_STACKSIZE=1G

1.1.4. Utilities

Tip

This utilities in this section is not provided on the XACS cloud computing platform.

1.1.4.1. Viewing VB orbitals: Moldendat

Viewing VB orbitals is available. To do that, you need to run a utility, called “moldendat”:

moldendat.exe MOfile vbdat [denfile] >&vbfile

where MOfile is an output file of Gaussian or GAMESS-US, or formatted Gaussian checkpoint file (.fchk); vbdat is a XMVB xdat file; if .fchk file is inputted, an optional XMVB density file with extension “.den” is also supported. The program will produce an NEW output file (vbfile) with the same format as input MO files, with which you can view VB orbitals with MOLDEN or MacMolPlt (for GAMESS-US only) packages.

1.1.4.2. Preparing integrals: PREINT

This utility is developed to prepare integrals and MO orbitals for XMVB. To run PREINT, simply type the command as following:

preint input >&output

where input is the input file (see below) and output is the user-defined output file. A typical input file for F2 molecule is shown below:

hf cc-pVTZ libcint spher
0 1
F 0.000000 0.000000 0.000000
F 0.000000 0.000000 1.400000

Here keyword libcint means the integrals will be generated by external library LIBCINT and spher means spherical integrals will be generated. Both keywords are optional. Currently “spher” can only be used with “libcint”.

The program provides three files for standalone XMVB jobs:

x1e.int containing 1-e integrals and MO orbitals
x2e.int storing 2-e integrals
INFO storing basis function information and coordinate of the molecule

The Basis sets and elements supported by current PREINT are:

STO-2G H-Ca,Sr
STO-3G H-Xe
STO-6G H-Kr
3-21G H-Cs
3-21G* H-Ar
3-21++G H-Ca
3-21++G* H-Ar
4-31G H-Cl
6-31G H-Zn
6-31G* H-Kr
6-31G** H-Zn
6-31+G H-Ca
6-31+G* H-Ca
6-31+G** H-Ca
6-31++G H-Ca
6-31++G* H-Ca
6-31++G** H-Ca
6-311G H-Ca,Ga-Kr,I
6-311G* H-Ca,Ga-Kr,I
6-311G** H-Ca,Ga-Kr,I
6-311+G H-Ca
6-311+G* H-Ca
6-311+G** H-Ca
6-311+G(2d,p) H-Ca
6-311++G H-Ca
6-311++G* H-Ca
6-311++G** H-Ca
6-311++G(2d,2p) H-Ca
cc-pVDZ H-Kr
cc-pVTZ H-Ca
aug-cc-pVDZ H-Kr
cc-pCVDZ H-Ca
cc-pCVTZ H-Ca
aug-cc-pCVDZ H-Ar
aug-cc-pCVTZ H-Ar
DZP H-Ba,La,Hf-Rn
TZP H-Ca

PREINT can also proceed DFT calculations. Currently supported DFT functionals are:

Exchange functionals: Slater, B88.
Correlation functionals: VWN1, VWN5, LYP.
Exchange-correlation functionals: SVWN1, SVWN5, BLYP.
Hybrid functionals: BHHLYP, B3LYP

More functionals will be implemented and supported in the future. All functionals support R-, U-, and RO-type calculations. To enable DFT calculations, just replace “hf” in the exmaple with functional names, for instance

ub3lyp cc-pVTZ libcint spher
0 1
F 0.000000 0.000000 0.000000
F 0.000000 0.000000 1.400000

1.1.4.3. Cartesian to spheric integral transformation: 6D25D

This utility transforms integrals from cartesian type to spheric (harmonic) type. Currently the utility supports D and F transformation only and not available for higher basis functions.

To run the utility, typing the command as following:

6d25d.exe [-if gau/gms/lib] [-of gau/std]

where option “-if” defines the sequential of cartesian F functions. Argument “gau” means the sequential in Gaussian and PREINT, “gms” means the sequential in GAMESS-US. Option and “lib” means the sequential by LIBCINT; -of” defines the output format of spheric F basis functions. Argument “gau” means the spheric F functions used in Gaussian package and “std” means standard spheric F function, which is different from the definition in Gaussian. By default, 6d25d will use Gaussian type for both input and output format.

After running 6d25d, the original cartesian integral files x1e.int, x2e.int and INFO will be overwritten by the spheric integrals. Make a backup of your cartesian integral files if you need them later.

1.1.4.4. Use NBOs as XMVB initial guess: NBOPREP

This utility read the NBOs obtianed from a previous GAMESS/Gaussian calculation, and transfer them to the XMVB readable formats so that user may use them as initial guess in later XMVB calculations with keyword GUESS=NBO.

The user need to run a GAMESS/Gaussian calculations with keyword

$NBO PLOT $END

to get files with name FILE.36 and FILE.37 which stores NBOs and PNBOs. Then run NBOPREP as following:

nboprep.exe outfile [NBO/PNBO]

where “outfile” refers to the output file of GAMESS/Gaussian program, and “NBO/PNBO” tells the program which kind of NBOs should be prepared for later XMVB calculation. The user may be able to use keyword GUESS=NBO by copying file “orb.nbo” generated by NBOPREP to the directory where the XMVB job will be proceeded.

1.1.4.5. Generate cube file for XMVB computation: vbcubegen

This utility generates cube grid file to visualize VB orbitals with other programs. It supports module distribution or stand-alone XMVB with keyword INT=CALC or INT=LIBCINT since basis function information is essential for generating grids. The syntax of this utility is

vbcubegen.exe xmofile

where xmofile refers to the XMO output file of the XMVB computation. After that, a cube grid file with the same file name as the xmo file will be generated, with which the user may visualize VB orbitals with programs such as GaussView, Multiwfn etc.

1.2. Input

The extension name of XMVB input file is “xmi”. All the contents is organized in sections and case insensitive. The input file is structured in sections with following rules:

The first line of an xmi file is the job title or description of the job and should not be replaced or omitted.
A section start with a line includnig only the section name and ends with a line with only “$END”.
All contents after “#” is recognized as a comment and will not be parsed.

Commonly used sections are:

Global control ($CTRL)
manual/input:VB structure description ($STR)
Fragments definition ($FRAG)
Orbital description ($ORB)
Initial guess description ($GUS)
Geometry description ($GEO)

An example of XMVB input file is shown below:

H2 L-VBSCF # Job title
$CTRL # Start of $CTRL section
NSTR=3 # 3 VB structures in this computation
NAO=2 NAE=2 ISCF=5 # VBSCF algorithm for RDM-based algorithm
IPRINT=3 # Printing level
ORBTYP=HAO FRGTYP=SAO # VB orbitals are HAO with symmetrized atomic orbital fragments (SAO)
INT=LIBCINT # Integrals are evaluated with LIBCINT library
BASIS=CC-PVTZ # Basis set is cc-pVTZ
GUESS=READ # Initial guess read from $GUS section
$END # End of $CTRL section
$STR # Start of $STR section for structure description with NSTR defined in $CTRL
1 2 # covalent structure H-H
1 1 # ionic structure H- H+
2 2 # ionic structure H+ H-
$END # End of $STR section
$FRAG # Start of $FRAG section since SAO fragment is requested
1*2 # 2 fragments, 1 atom included in each
SPZDXXDYYDZZ 1 # Fragment 1, basis functions s, pz, dxx, dyy and dzz on atom 1
SPZDXXDYYDZZ 2 # Fragment 2, basis functions s, pz, dxx, dyy and dzz on atom 2
$END # End of $FRAG section
$ORB # Start of $ORB section for orbital description
1*2 # 2 VB orbitals, each includes 1 fragment (since fragments defined in $FRAG)
1 # orbital 1, with only fragment 1
2 # orbital 2, with only fragment 2
$END # End of $ORB section
$GEO # Start of $GEO section since INT=LIBCINT requested
H 0.0 0.0 0.0     # H2 coordinate given in Cartesian
H 0.0 0.0 0.74
$END # End of $GEO section
$GUS # Start of $GUS section since GUESS=READ requested, pasted from previous computation result
15 15
# ORBITAL 1 NAO = 15
-0.3532245024 1 -0.5363311264 2 -0.2343104477 3 -0.0000000000 4
 0.0000000000 5 -0.0199961314 6 -0.0000000000 7  0.0000000000 8
-0.0192894825 9 -0.0003896018 10 0.0000000000 11 -0.0000000000 12
-0.0003896018 13 0.0000000000 14 -0.0020820012 15
# ORBITAL 2 NAO = 15
 0.3532245024 16 0.5363311264 17 0.2343104477 18 0.0000000000 19
-0.0000000000 20 -0.0199961314 21 0.0000000000 22 -0.0000000000 23
-0.0192894825 24 0.0003896018 25 -0.0000000000 26 -0.0000000000 27
0.0003896018 28 0.0000000000 29 0.0020820013 30
$END

1.2.1. Global control ($CTRL)

The $CTRL section contains the information of how a job is performed. The input format is name=value or name=option, except for the keywords which need no values or options. <enter> and <space> are used to separate keywords. If a keyword accepts several options in a time, the options are separated with “,”.

1.2.1.1. Keywords for Global Control

1.2.1.1.1. BPREP

This keyword initiates an integral transformation from primitive basis functions to VB basis functions with $BFI (see our offline manual) needed. The transformation may freeze core orbitals, remove some primitive basis functions which are not involved in VB calculation, and hybridize basis functions. XMVB will use primitive basis functions without transformation if this keyword is missing.

Note

This keyword cannot be used together with ORBTYP=HAO or GUESS=MO (see descriptions for keywords ORBTYP=option and GUESS=option)

1.2.1.1.2. EPG=n

Set the convergent criterion of energy to $10^{-n}$. Default value is 7.

1.2.1.1.3. GPG=n

Set the convergent criterion of gradient. Floating point number is inputted. Default value is 2.D-3 for ISCF=1/3, 1.D-3 for ISCF=2/5, and 1.D-4 for ISCF=6.

1.2.1.1.4. ITMAX=n

n is the maximum number of iterations. Default value is 200.

1.2.1.1.5. NMUL=n

n is the spin multiplicity (2S + 1) of system. Default value is 1, which means singlet state.

1.2.1.1.6. NAO=m

m is the number of active VB orbitals whose occupation number varies in the structures. NAO is required if keywords STR or ISCF=5 (see below) is specified.

1.2.1.1.7. NAE=n

n is the number of active VB electrons which occupy the active orbitals. NAE is required if keywords STR or ISCF=5 (see below) is specified.

1.2.1.1.8. NSTR=n

n is the number of VB structures (or determinants). This keyword can be omitted if STR (see below) is assigned.

1.2.1.1.9. STR=options

This keyword generates VB structures automatically and hence NSTR and the $STR section are not needed. This keyword requires NAO and NAE to declare the active space. Users may use one or several of the following options:

COV: Covalent structures will be generated.
ION[(n-m)]: Ionic structures will be generated. A simple ION will generate all ionic structures; ION(n,m) will generate only the $n^\textrm{th}$ and $m^\textrm{th}$ order ionic structures and ION(n-m) will generate ionic structures from the $n^\textrm{th}$ to the $m^\textrm{th}$ order.
FULL: All VB structures will be generated.

1.2.1.1.10. FIXC

Request to fix structure coefficients for VB structures. In VB theory, the coefficients are obtained by solving the secular equation

\[\mathbf{HC} = E\mathbf{MC}\]

For some special purposes, one may want to fix the coefficients. In such situation, the coefficients are inputted following the corresponding VB structures and the energy will be obtained directly by

\[E=\frac{\sum_K\sum_LC_KC_LH_{KL}}{\sum_K\sum_LC_KC_LM_{KL}}\]

For example, the following input will constrain the coefficients of the three VB structures to be 1.0:0.5:0.5

$STR
1 2 1.0
1 1 0.5
2 2 0.5
$END

The corresponding wave function will in the expression

\[\Psi = N\left( S_1 + 0.5S_2 + 0.5S_3 \right)\]

where N is the normalization coefficient.

1.2.1.1.11. GROUP=EXP

Divide VB structures into groups according to the expression EXP. An expression with n structures divided into m groups can be expressed as:

\[\ldots , S_{i1},\ldots ,, \ldots , S_{j2},\ldots,,\ldots,,,\ldots,S_{nm}\ldots\]

Here $S_{i1} \ldots S_{nm}$ are the structure numbers, a comma “,” is used to separate the structures numbers in the same group, and two commas “,,” is used to separate different groups. Coefficients of structures should be given in Global control ($CTRL), similar to FIXC. The ratio of VB structures within the same group will be fixed, as introduced in FIXC. The coefficients of VB structures in different groups will not be fixed and shall be optimized by solving secular equation. Following is an example:

$CTRL
NSTR=3
GROUP=1„2,3
$END
$STR
1 2 1.0 # S1
1 1 0.5 # S2
2 2 0.5 # S3
$END

The above example devide 3 VB structures into 2 groups:

Group 1. $G_1 = S_1$
Group 2. $G_2 = 0.5(S_2 + S_3)$

Hence a 3 structure problem becomes a 2 “structure” problem:

\[\Psi = C_1G_1 + C_2G_2\]

where $C_1$ and $C_2$ are coefficients of $G_1$ and $G_2$ obtained by solving secular equation. The finalwave function can be expressed as

\[\Psi = C_1S_1 + \frac{C_2}{2}S_2 + \frac{C_2}{2}S_3\]

1.2.1.1.12. NSTATE=n

Energy, coefficients and weights of structures for the $n^\textrm{th}$ excited state, rather than for the ground state, will be calculated and printed out. The values of n can be:

0: The ground state.(Default)
n: The $n^\textrm{th}$ excited state.

Note

VB orbitals are optimized by minimizing the energy of required state. When the $n^\textrm{th}$ excited state is requested, the $(n+1)^\textrm{th}$ root will be chosen as the $n^\textrm{th}$ excited state when solving the secular equation. Thus, n must be smaller than the number of structures.
For VBCI calculaitons, NSTATE can be only 0 or 1.

1.2.1.1.13. IPRINT=n

Printing levels for XMVB. The available levels are:

1: Initial guess, energy, coefficients, weights, and orbitals will be printed. This is the default printing level.
2: All contents in IPRINT=1, Hamiltonian and overlap matrices in terms of VB structures, and population analysis will be printed.
3: All contents in IPRINT=2, density matrix and orbital overlap matrix will be printed.

1.2.1.1.14. SORT

Sort the VB structures in descending order according to coefficients.

1.2.1.1.15. CTOL=tol

Set the Coefficient TOLerance when printing coefficients and weights of VB structures.

Only the coefficients and weights of VB structures whose absolute values of coefficients are not smaller than tolerance tol will be printed. The default tolerance is 0, which means all structures will be printed.

Note

The tolerance tol is a real parameter. For instance,

CTOL=0.01

means that only structures whose absolute values of coefficients larger than or equal to 0.01 will be printed. For VBCI this keyword is not functioning

1.2.1.1.16. CICUT=n

Set cut threshold to $10^{-n}$ for CI configurations. The contribution of a CI configuration is estimated by perturbation theory. If the contribution is less than the threshold, the configuration will be discarded. This will reduce the computational effort for CI calculations. Recommended values are 5 or 6. Default value is 0 (no cut).

1.2.1.1.17. NCOR=m

In VBCI or VBPT2 calculations, the first m orbitals (2m electrons) will be frozen in the VBCI or VBPT2 calculation. In BOVB caluclations, the first m orbitals will be kept as VBSCF orbitals. The default value is 0, which means all orbitals will be counted in VBCI, VBPT2 or BOVB.

1.2.1.1.18. GUESS=option

This keyword describes the way to generate or read the initial guess for a VB computation.

Valid options can be:

AUTO: The program automatically provides guess orbitals by diagonalizing a fragmant-localized Fock matrix. This is the default option.
UNIT: The first basis function of an orbital in $ORB is set to be the guess for the orbital.
NBO: Initial guess will be obtained from NBOs.
READ: Guess orbitals are read from external file, which should be provided by user. MO: Initial guess of VB orbitals will be obtained directly from molecular orbitals.
RDCI: Initial guess in VBCI type will be given by users.

Note

GUESS=MO cannot be used with BPREP. GUESS=NBO cannot be used with BPREP and needs an extra preparation by NBOPREP. GUESS=AUTO cannot be used when some orbitals contain only one basis function (see Orbital description ($ORB) section).

1.2.1.1.19. WFNTYP=option

Options for the way to expand the many-electron wave functions of system.

STR: VB structures are used. (Default)
DET: VB determinants are used for state functions, instead of VB structures.

1.2.1.1.20. VBFTYP=option

Options for the way to expand VB structures.

PPD: paired-permanent-determinant algorithm is used.
DET: Slater determinant algorithm is used.

By default, the program will decide which one to use according to the system, method, or algorithm the users choose.

Note

PPD expansion can be used only with ISCF=1 or ISCF=3.
ISCF=5, VBPT2, VBCI, DFVB, solvation VB methods, DEN, and IPRINT $\ge$ 2 will use DET expansion automatically.
All systems with multiplicity larger than 2 will be calculated with DET expansion.
Systems with electrons in VB calculation larger than 14 will be calculated with DET expansion.

1.2.1.1.21. ORBTYP=option

Specify the type of VB orbitals. Valid options are:

HAO: Hybrid Atomic Orbitals are used.
BDO: Bond Distorted Orbitals are used.
OEO: Overlap Enhanced Orbitals are used.
GEN: VB orbitals are defined by users. (Default)

Note

Fragments definition ($FRAG) is needed if ORBTYP=HAO is specified. The Fragments definition ($FRAG) section will specify the fragments based on atoms or basis functions and orbitals will be assigned in Orbital description ($ORB) section based on the fragment definitions in Fragments definition ($FRAG).
ORBTYP=OEO does not need Fragments definition ($FRAG) and Orbital description ($ORB) sections since the OEOs are delocalized in the whole system.
ORBTYP=GEN does not need Fragments definition ($FRAG) section, and orbitals will be described in terms of VB basis functions explicitly in Orbital description ($ORB) section.
ORBTYP=HAO cannot be used with BPREP.
ORBTYP=BDO can be used with other orbital types, such as ORBTYP=HAO,BDO. ORBTYP=BDO is equivalent to ORBTYP=GEN,BDO.

1.2.1.1.22. FRGTYP=option

Specify the type of fragments when ORBTYP=HAO.

ATOM: The fragments of system will be defined with atoms. This is the default.
SAO: The fragments of system will be defined with symmetrized atomic orbitals.

Note

Fragments definition ($FRAG) is required for FRGTYP=SAO. For FRGTYP=ATOM, each atom is considered as a fragment if no FRAG section appears in the input file.

1.2.1.2. Keywords for Computational Methods and Algorithms

1.2.1.2.1. HF/RHF/UHF/ROHF

A Hartree-Fock calculation will be proceeded. RHF/UHF/ROHF represent the restricted, unrestricted and restricted open-shell Hartree-Fock calculations respectively. When only “HF” is assigned, RHF will be proceeded when system is singlet and UHF for other cases.

1.2.1.2.2. Density Functional Theory

A DFT calculation will be proceeded. Currently supported keywords and corresponding functionals are listed below:

LDA Functionals
1. Slater Slater exchange functional
2. VWN/VWN5 Vosko-Wilk-Nusair correlation functional
3. VWN1 Another Vosko-Wilk-Nusair correlation functional
4. SVWN/SVWN5 Slater + VWN5 XC functional
5. SVWN1 Slater + VWN1 XC functional
GGA Functionals
1. B88 Beck88 exchange fuctional
2. LYP Lee-Yang-Parr correlation functional
3. BLYP Becke88 + Lee-Yang-Parr XC functional
Hybrid Functionals
1. BHHLYP 0.5 B88 + 0.5 HFX + LYP hybrid functional
2. B3LYP Becke’s 3 parameter hybrid functional, with VWN1 involved
3. B3LYP5 Becke’s 3 parameter hybrid functional, with VWN5 involved

The users may use “R”, “U”, and “RO” ahead of the name of functional to specify restricted, unrestricted or restricted open-shell calculations, the same as HF method. For example, “RB3LYP” will run the restricted B3LYP calculation. If only the name of functional is specified, restricted calculation will be run for singlet and unrestricted for others.

1.2.1.2.3. VBSCF

A VB Self-Consistent Field computation is requested. This is the default method for the XMVB program.

1.2.1.2.4. BOVB

Ask for a Breathing Orbital VB (BOVB) calculation.

Note

BOVB method cannot be used with VBCI.

BOVB method is usually more difficult to converge than VBSCF. Thus, it is recommended to run a BOVB job with a good initial guess. It is recommended to run a VBSCF calculation first, followed by the BOVB calculation with optimized VBSCF orbitals as the initial guess.

1.2.1.2.5. ABOVB

Ask for an Approximate Breathing Orbital VB (A-BOVB) calculation.

1.2.1.2.6. BLW

Block Localized Wavefunction (BLW) method is requested. With this keyword specified, Global control ($CTRL) will not be read and the structure will be generated automatically. The users only need specify the type of VB orbitals (see FRGTYP=option and ORBTYP=option above).

Note

The implementation of the BLW method in the program is not optimized. Users are recom mended to run GAMESS-BLW for a BLW calculation.

1.2.1.2.7. VBCIS:

Ask for a VBCIS calculation.

1.2.1.2.8. VBCISD

Ask for a VBCISD calculation.

1.2.1.2.9. VBCIDS

Ask for a VBCIDS calculation.

1.2.1.2.10. VBPT2

A VBPT2 computation will be performed.

1.2.1.2.11. DFVB

Ask for a DFVB calculation.

1.2.1.2.12. SCC

Size-Consistent Correction in DFVB computations will be switched on.

1.2.1.2.13. VBEFP

Ask for a VBEFP calculation.

1.2.1.2.14. VBPCM

Ask for a VBPCM calculation.

1.2.1.2.15. VBEFPPCM

Ask for a VBEFP/PCM calculation.

1.2.1.2.16. TBVBSCF

Activate tensor-based VBSCF. Currently TBVBSCF is valid only when:

ISCF=5, NAO=m and NAE=n are selected.
Structures are generated automatically with STR=options.
Number of active electrons should be at least 4, in which 2 for both $\alpha$ and $\beta$ parts.

1.2.1.2.17. VMAX=n

The maximum number of $\sigma$ kept in Davidson diagonalization. The default value is 10. Only for TBVBSCF.

1.2.1.2.18. READCOEF

Read a file “coef” (see File with coefficients for the structures/determinants (.coeff)) with coefficients of the first n structures as the initial guess of Davidson diagonalization. The file may be obtained from a previous TBVBSCF. Only for TBVBSCF.

1.2.1.2.19. ISCF=n

ISCF specifies orbital optimization algorithm. The value n currently can be:

1: Numerical gradients with forward-difference approximation are used with the DFP-BFS algorithm. This is the default option of XMVB.
2: Analytical gradients in terms of basis functions with the L-BFGS algorithm. This algorithm involves only the first-order density matrix and is not suitable in cases displaying structures that are orthogonal to each other.
3: Numerical gradients with central-difference approximation are used with the DFP-BFS algorithm.
4: Optimization with Generalized Brillium Theory(GBT) is requested.
5: Analytical gradients in terms of VB orbitals with the L-BFGS algorithm. This is the most efficient algorithm so far. This option needs NAO and NAE.
6: VBSCF with full hessian matrix. NAO and NAE are needed for this option. This algorithm is potentially faster and more robust than ISCF=5, but it is still under development and thus is not recommended in the current version of the program.

1.2.1.2.20. EIGMTHD=option

Specify the way to solve the secular equation and get the energy and coefficients. The available options can be:

FULL: Solving secular equation in the traditional way.
ITER: An iterative algorithm will be used to solving secular equation. This will be helpful for VBCI or VBSCF with large number of structures.

1.2.1.2.21. WSTATE=EXP

Activate the state-average VBSCF calculation for ISCF=1-5. WSTATE may provide an array containing non-zero weights of the specific states. Following is the example for

\[E = 0.5E_3 + 0.3E_5 + 0.2E_8\]

$CTRL
NSTR=10 WSTATE(3)=0.5,0.0,0.3,0.0,0.0,0.2
$END

1.2.1.3. Keywords for Integrals

1.2.1.3.1. INT=option

Read integrals from file or calculate them directly. The valid options can be:

READ: Read integrals from existing file “x1e.int” and “x2e.int”. This is the default option.
CALC: Calculate integrals directly. Section Geometry description ($GEO) is essential.
LIBCINT: Integrals are calculated directly by an external library LIBCINT. Section Geometry description ($GEO) is essential.

1.2.1.3.2. BASIS=basis_set

Assigning the basis set when INT=CALC is requested. Basis sets are expressed the same way as Gaussian, i.e. 6-31G*, aug-cc-pVTZ etc. The supported basis sets can be found in Preparing integrals: PREINT.

1.2.1.3.3. NCHARGE=n

Charge of the system in current XMVB calculation. Default is 0, which means the neutural system. Positive numbers denote a cation system and negative numbers mean the system is anion. This keyword will also specify the number of electrons in current calculation, NEL is not needed anymore in such case.

1.2.1.3.4. ERI=CD

This keyword activates the Cholesky decomposotion for ERIs. Only valid with ISCF=5 and INT=CALC.

1.2.1.3.5. CDTOL=float

The tolerance of Cholesky decomposition. Default is $1 \times 10^{-10}$. Float may be expressed like 1.d-6, 0.001 etc.

1.2.1.4. Keywords for Wave Function Analysis

1.2.1.4.1. BOYS

Boys localization is requested for the final VB orbitals.

Note

It is strongly recommended to use this keyword for VBSCF. This makes VB orbitals easier to be interpreted and more physically meaningful.

Boys localization is available only for VBSCF method.

Boys localization can be only used in cases in which orbitals are separated into blocks, and there is no common basis function between blocks.

1.2.1.4.2. DEN

First-order density matrix is requested. The result will be written to a file with extended name “den”.

1.2.1.4.3. OUTPUT=AIM

WFN file for AIM2000 program will be printed. This is available only in module distribu- tion. A $AIM with WFN filename is relevant for this keyword. Without $AIM, the content of WFN file will be stored in “.dat” file of GAMESS-US.

1.2.1.5. Keywords for Symmetry

1.2.1.5.1. SYMM=option

Options for the point group used in VB computation. Currently only Abelian groups are supported. More details are shown in the table.

Note

The main axis of the molecule should be put along the Z-axis when $C_\textrm{2}$, $C_\textrm{2v}$, $C_\textrm{2h}$, $D_\textrm{2}$and $D_\textrm{2h}$ are applied.

This keyword cannot be used together with OEO type of VB orbitals(see keyword ORBTYP=option) and MO type of initial guess.

1.2.1.5.2. IRRP=option

Options for the irreducible representation used in VB computation. Valid options are shown in the table.

Table 1.2.1 Supported point groups and irreducible representations, and corresponding options for keywords
SYMM Option	Point Group	IRRP Option	Irreducible Representation
CS	$C_\textrm{s}$	AP, APP	$A'$, $A''$
CI	$C_\textrm{i}$	AG, AU	$A_\textrm{g}$, $A_\textrm{u}$
C2	$C_\textrm{2}$	A, B	$A$, $B$
C2V	$C_\textrm{2v}$	A1, A2, B1, B2	$A_\textrm{1}$, $A_\textrm{2}$, $B_\textrm{1}$, $B_\textrm{2}$
C2H	$C_\textrm{2h}$	AG, AU, BG, BU	$A_\textrm{g}$, $A_\textrm{u}$, $B_\textrm{g}$, $B_\textrm{u}$
D2	$D_\textrm{2}$	A1, A2, B1, B2	$A_\textrm{1}$, $A_\textrm{2}$, $B_\textrm{1}$, $B_\textrm{2}$
D2H	$D_\textrm{2h}$	AG, AU, B1G, B2G, B3G, B1U, B2U, B3U	$A_\textrm{g}$, $A_\textrm{u}$, $B_\textrm{1g}$, $B_\textrm{2g}$, $B_\textrm{3g}$, $B_\textrm{1u}$, $B_\textrm{2u}$, $B_\textrm{3u}$

1.2.1.6. Keywords for Previous Version

The following keywords appear in the previous version and are not available since version 2.0. This part is important only for those who are used to the previous version.

1.2.1.6.1. CIG

This keyword has been modified as GUESS=RDCI.

1.2.1.6.2. DET

This keyword has been replaced by WFNTYP=DET.

1.2.1.6.3. EXC

This keyword has been replaced by NSTATE=1.

1.2.1.6.4. IOPT=n

This keyword has been replaced by keyword ISCF=n.

1.2.1.6.5. IOUT=n

1.2.1.6.6. NODIIS

1.2.1.6.7. VBXM=n

This keyword has been replaced by keywords WFNTYP=option and VBFTYP=option.

1.2.2. Required for BPREP($BFI)

The BFI section specifies how to transform primitive basis function to VB basis functions.The primitive basis functions are those used in GAMESS, Gaussian or PREINT and VB basis functions are used in XMVB. The Syntax of $BFI section is:

$BFI
NFROZ NBAS
List of frozen MOs
List of basis functions
$END

Here NFROZ is the number of frozen MOs and NBAS is the number of VB basis functions used in XMVB. Then frozen MOs and basis functions will be listed respectively. If there is no MO to be frozen, place a blank line there. The VB basis functions may be reordered according to how users list them. This new order will be used in $ORB section for the orbital description VB structure.

Following are two examples for the BFI section. The first example comes from the HF molecule with 6-31G basis set, where basis functions are not hybridized:

$BFI
3 6
1 4 5
1 2 4 7 8 11
$END

In this example, three MOs 1, 4 and 5 are frozen and 6 VB basis functions are kept for the XMVB calculation. Primitive basis functions 3, 5, 6, 9 and 10 are removed from the list as the corresponding MOs are frozen. Note that the fourth VB basis function is the primitive basis function 7 according to the list, not primitive basis function 4 anymore. The second example comes from the CH4 molecule with 6-31G basis set, showing the orbital freezing and the hybridization of basis functions:

$BFI
8
3 4
1 1 1 1 1 3 3
0 1
0 2
0 4
0 7
0 8
0 11
5 12 0.5 14 0.5 16
5 13 0.5 15 0.5 17
$END

Here MOs 1, 3 and 4 are frozen and 8 hybrid VB basis functions are used in XMVB calculation. Line “1 1 1 1 1 1 3 3” specifies the number of primitive basis function in each VB basis function. Following lines specifies how the VB basis functions are hybridized. In this example,the 7-th VB basis function is a hybrid basis function composed of 3 primitive basis functions 12, 14 and 16

1.2.3. description ($STR)

The $STR section describes the information of VB structures or VB determinants if DET of $CTRL section is specified. For VB structures, paired electrons, which may be lone pairs or covalent bonds, should be written first followed by unpaired electrons. The number of unpaired electrons depends on the spin multiplicity. For example: For a structure with three lone pairs (orbitals 1, 2, and 3), one covalent bond (orbitals 4 and 5), and one unpaired electron (orbital 6), the structure is expressed as,

1 1 2 2 3 3 4 5 6

For determinants, all alpha orbitals are listed first, followed by beta orbitals. For example: A determinant of alpha orbtials 1, 2, 3, 4, and 6 and beta orbtials 1, 2, 3, and 5 is expressed as

1 2 3 4 6 1 2 3 5

Note that it is strongly recommended to write the most important structure as the first one.

This can avoid potential problems in VBCI.

If BOVB is specified in $CTRL section, the program will try to convert the VB orbitals into breathing orbitals. It uses automatically different orbitals for different structures. For example: If the initial VB structures are:

1 2 3
1 2 4
1 3 5

The program will convert them to:

1 2 3
6 7 4
8 9 5

Note that the VB structures should be independent. VB structures are recommended to be written in the following orders:

Inactive Active

where “Inactive” stands for the inactive orbitals which keep doubly occupied in all structures; “active” stands for the active orbitals whose occupation varies in the structures. The singly occupied orbitals in high-spin systems should always be put in the tail of the structures.

Following are the examples of typical bonding patterns and their corresponding manual/input:VB structure description ($STR) and Global control ($CTRL) sections, in which only active orbitals are labeled:

System of 2-electrons on 2-centers

$CTRL
nstr=3 nmul=1
$END
$STR
1 2 ; S1
1 1 ; S2
2 2 ; S3
$END

System of 3-electrons on 2-centers

$CTRL
nstr=2 nmul=2
$END
$STR
1 1 2 ; S1
2 2 1 ; S2
$END

System of 3-electrons on 3-centers

$CTRL
nstr=8 nmul=2
$END
$STR
1 2 3 ; S1
2 3 1 ; S2
1 1 3 ; S3
3 3 1 ; S4
2 2 3 ; S5
2 2 1 ; S6
1 1 2 ; S7
3 3 2 ; S8
$END

System of 4-electrons and 3-centers

6 VB structures (3 VB orbitals with 4 electrons, singlet)

$CTRL
nstr=6 nmul=1
$END
$STR
1 1 2 3 ; S1
1 1 2 2 ; S2
1 1 3 3 ; S3
1 2 3 3 ; S4
2 2 3 3 ; S5
2 2 1 3 ; S6
$END

1.2.4. Fragments definition ($FRAG)

Generally, the $FRAG section is required if ORBTYP=HAO. In this section, fragments in which VB orbitals are localized will be defined and the orbitals will be generated with the basis functions specified in the fragments.

The syntax of $FRAG is:

$FRAG
nf(1), nf(2), . . . nf(N)
[basis function description(1)] lf(1,1), lf(2,1), . . . lf(nf(1),1)
[basis function description(2)] lf(1,2), lf(2,2), . . . lf(nf(2),2)
. . .
[basis function description(N)] lff(1,N), lf(2,N), . . . lf(nf(N),N)
$END

Here the system is separated into N fragments. nf(i) means the number of atoms or basis functions in the $i^\textrm{th}$ fragment, and lf(j,i) is the atom or basis function j in the $i^\textrm{th}$ fragment. Basis function description is needed only when FRGTYP=SAO is chosen. Following is an example of H₂ molecule with FRGTYP=ATOM:

$CTRL
NSTR=3 ORBTYP=HAO
FRGTYP=ATOM
$END
$STR
1 2
1 1
2 2
$END
$FRAG
1 1
1
2
$END
$ORB
1 1
1
2
$END

The above $FRAG specifies two fragments, where one atom is in each fragment. Fragment 1 includes the first H atom and fragment 2 includes the second H atom. With this definition, users only need to specify fragment in which an orbital is located in Orbital description ($ORB) section. With FRGTYP=SAO, the fragments are specified by the type of basis functions. Following is an example of HF molecule with 6-31G basis set:

$CTRL
NSTR=3 VBFTYP=DET DEN
ISCF=5 NAO=2 NAE=2
ORBTYP=HAO FRGTYP=SAO
$END
$STR
1:4 5 6
1:4 5 5
1:4 6 6
$END
$FRAG
1 1 1 1
S 1
SPZ 2
PX 2
PY 2
$END
$ORB
1 1 1 1 1 1
2
2
3
4
2
1
$END

For the second fragment, “1” in the first line of $FRAG means that the block contains basis functions located on one atom; “SPZ 2” means that the fragment includes the $s$ and $p_z$ basis functions in the second atom. The basis functions are described by groups of $s$, $p$, $d$, $f$, etc. For example, a fragment including $s$, $p_z$, $d_{xx}$, $d_{yy}$, $d_{zz}$, $f_{zzz}$, $f_{xxz}$ and $f_{yyz}$ basis functions in atoms 1 and 2 should be described as

$FRAG
2
spzdxxyyzzfzzzxxzyyz 1 2
$END

or

$FRAG
2
spzdxxdyydzzfzzzfxxzfyyz 1 2
$END

Here “s” means basis function $s$, “pz” means basis function $p_z$, “dxxyyzz” means $d_{xx}$, $d_{yy}$ and $d_{zz}$, and “fzzzxxzyyz” means $f_{zzz}$, $f_{xxz}$ and $f_{yyz}$. The ordering of basis functions are not compulsively defined, but the basis functions with the same type of $s$, $p$, $d$ and $f$ should be written together. For example, the above description can be written equivalently as

$FRAG
2
spzfzzzfxxzfyyzdxxdyydzz 1 2
$END

or

$FRAG
2
spzfxxzfyyzfzzzdxxdyydzz 1 2
$END

as users like.

1.2.5. Orbital description ($ORB)

Required When ORBTYP=HAO or ORBTYP=GEN.

The first line describes the number of basis functions (or fragments) that are used for VB orbitals. For instance, max(i) means that the $i^\textrm{th}$ orbital is expanded as max(i) functions (fragments), which are specified in the following lines. If the value of max(i) is 1, it means that the corresponding orbital will not optimized. From the second line, the indices of basis functions are listed, where one orbital begins with one new line. Following is example:

4 2
4 5 6 ; orbital 1 is expanded with 4 basis functions (fragments)
3 5 6 ; orbital 2 is expanded with 4 basis functions (fragments)
2 ; orbital 3 is expanded with 2 basis functions (fragments)

Note

It is important to emphasize again that the $n^\textrm{th}$ VB basis function in $ORB section is NOT necessarily the $n^\textrm{th}$ primitive basis function, but the $n^\textrm{th}$ VB basis function specified in the BFI section.
It is suggested to write the most important basis function as the first one, as the program takes the first function as the “parent” function for the orbital if GUESS=UNIT. This can avoid potential problems in convergence.
If ORBTYP=OEO is chosen, the $ORB is not needed. All the orbitals will be delocalized in the whole system, which means orbitals will use all basis functions.
If the users want to freeze (not optimize) some orbitals in the calculation, simply assigning the number of basis functions (fragments) of the corresponding orbital to “0”. For example, “0*5 2 2” means that there are totally 7 VB orbitals and the first 5 will be frozen during SCF iterations. In this case, an initial guess should be provided either by GUESS=READ or GUESS=MO.

1.2.6. AIM Section($AIM)

This section is relevant if OUTPUT=AIM is specified. The content of this section is an optional file name specified by users. This file name will be used as the WFN file name. By default, the content of WFN file will be stored in ".wfn" file with the same name as input for standalone distribution and in ".dat" file for module distribution.

1.2.7. Initial guess description ($GUS)

Required when GUESS=MO or GUESS=READ or GUESS=RDCI.

When GUESS=MO is required in manual/input:VB structure description ($STR) section, $GUS describes how VB orbital guess comes from MOs. An example of $GUS from H₂ calculation is shown below:

$GUS
1 1
2 1
$END

The example shows that both VB orbitals 1 and 2 will get the initial guess from MO 1. All orbitals should be specified in this section.

If GUESS=READ or GUESS=RDCI is required, orbitals from previous computation will be read from $GUS section. Thus, this section now contains the orbitals provided as the initial guess. The content is the same as the ORB file of the previous computation. See File with optimized VB orbitals (.orb) and orbital guess (.gus) for details of the content.

1.2.8. Required when SCF=n ($SCF)

The section contains n columns of structure coefficients, each denotes a state. Following is the example for SCF=2 with 2 structures:

$SCF
1.0 1.0
2.0 -2.0
$END

Thus the result of $\langle S_1+2S_2|\hat{H}|S_1-2S_2\rangle$will be calculated, where $\ S_1$and $\ S_2$denote the 2 structures.

1.2.9. Geometry description ($GEO)

Required when INT=CALC or INT=LIBCINT.

section contains the geometry of the system in cartesian coordinates, and the unit is Angstrom. Both Gaussian and GAMESS-US format are supported. Here both examples of the same geometry are given:

Gaussian Format:

$GEO
F 0.0 0.0 -0.7
F 0.0 0.0 0.7
$END

GAMESS-US Format:

$GEO
F 9.0 0.0 0.0 -0.7
F 9.0 0.0 0.0 0.7
$END

The users may choose their favorite.

1.3. Output

1.3.1. Main XMVB output file (.xmo)

The output of XMVB is stored in a file with extesion “xmo”. The following is an example for stand-alone XMVB:

    *************************************************************

         M   M    MM MM    M   M    MMMM        MMMM      M
          M M     M M M    M   M    M   M          M      M
           M      M M M     M M     MMMM   MMM  MMMM      M
          M M     M   M     M M     M   M       M         M
         M   M    M   M      M      MMMM        MMMM  M   M

    *************************************************************

                    Released on Jun 14, 2015

    Cite this work as:

    (a) Z. Chen, F. Ying, X. Chen, J. Song, P. Su, L. Song, Y.
    Mo, Q. Zhang and W. Wu, Int. J. Quantum. Chem., 2015, 115,
    737 (b) L. Song, Y. Mo, Q. Zhang, W. Wu, J. Comput. Chem.
    2005, 26, 514.


 Job started at Mon Jun  8 14:36:17 2015
 Work Directory at /home/fmying/VB_Workshop/Tutorial/lesson1/ex1   PID =       18199
---------------Input File---------------
H2 VBSCF FRAG BY ATOM
$ctrl
str=full nao=2 nae=2 # generate all VB structures with 2 active orbitals and 2 active electrons
orbtyp=hao frgtyp=atom # Construct VB orbitals with HAOs, fragmented by atom
iscf=5 # VBSCF algorithm with reduced density matrix
iprint=3 # Full print level
itmax=2000 # Maxinum number of iterations is set to 2000.
$end
$frag
1 1 # 2 fragments, each contains 1 atom
1 # First fragment, containing atom 1
2 # Second fragment, containing atom 2
$end
$orb
1 1 # 2 orbitals, each contains 1 fragment
1 # Orbital 1, containing fragment 1
2 # Orbital 2, containing fragment 2
$end
 ---------------End of Input--------------
Number of   0th ion  structures  is:          1  from          1 to          1
Number of   0th ion determinants is:          2  from          1 to          2
Number of   1th ion  structures  is:          2  from          2 to          3
Number of   1th ion determinants is:          2  from          3 to          4
Total number of  structures  is:          3
Total number of determinants is:          4


 READING INTEGRALS...

 Reading 2-e Integrals...
 Done

 OPTIMIZATION METHOD: LBFGS WITH ANALYTICAL ORBITAL GRAD IENT

 Number of Structures:    3

 The following structures are used in calculation:

     1 *****    1  2
     2 *****    1  1
     3 *****    2  2

 Nuclear Repulsion Energy:       0.715104

 Diagonalize Fock Matrix...


 --------------Initial Guess--------------
   5   5
 0.5388052224   1   0.5354436033   2   0.0000000000   3   0.0000000000   4
 0.1668081998   5
 0.5388052224   6   0.5354436033   7   0.0000000000   8   0.0000000000   9
-0.1668081998  10
 --------------End of Guess--------------

 VBDET is applied

                   10  Coefficients                    10  Independent

                ITER           ENERGY               DE              GNORM
                  1         -1.0806051993     -1.0806051993      0.3355377748
                  2         -1.0995681133     -0.0189629140      0.2387997424
                  3         -1.1308761698     -0.0313080566      0.1539314497
                  4         -1.1420399230     -0.0111637531      0.1093622053
                  5         -1.1465763955     -0.0045364726      0.0120926244
                  6         -1.1466054202     -0.0000290247      0.0001989203

                        VBSCF converged in     6 iterations

 Total Energy:      -1.14660543

 First Excited:     -0.256277


 The Last Change in Energy:  -0.000000

 Number of Iteration:    6


                 ******  MATRIX  OF  OVERLAP  ******


            1          2          3
   1     1.000000   0.820727   0.820727
   2     0.820727   1.000000   0.507832
   3     0.820727   0.507832   1.000000



               ******  MATRIX  OF  HAMILTONIAN ******


            1          2          3
   1    -1.857032  -1.547823  -1.547823
   2    -1.547823  -1.558228  -1.080145
   3    -1.547823  -1.080145  -1.558228



              ******  COEFFICIENTS OF STRUCTURES ******

     1     0.83675  ******    1  2
     2     0.09850  ******    1  1
     3     0.09850  ******    2  2



              ******  COEFFICIENTS OF DETERMINANTS ******

                              a
                              b

     1     0.48184  ******    2
                              1
     2     0.48184  ******    1
                              2
     3     0.09850  ******    1
                              1
     4     0.09850  ******    2
                              2



              ******  WEIGHTS OF STRUCTURES ******

     1     0.83545  ******    1  2
     2     0.08228  ******    1  1
     3     0.08228  ******    2  2

         Lowdin Weights

     1     0.53757  ******    1  2
     2     0.23121  ******    1  1
     3     0.23121  ******    2  2

         Inverse Weights

     1     0.94072  ******    1  2
     2     0.02964  ******    1  1
     3     0.02964  ******    2  2



                 ******  OPTIMIZED ORBITALS  ******


              1          2
   1     0.763386   0.000000
   2     0.307544   0.000000
   3     0.000000   0.000000
   4     0.000000   0.000000
   5     0.032894   0.000000
   6     0.000000   0.763386
   7     0.000000   0.307544
   8     0.000000   0.000000
   9     0.000000   0.000000
  10     0.000000  -0.032894



       ******  ORBITALS IN PRIMITIVE BASIS FUNCTIONS ******


                    1          2
    1        S    0.763386   0.000000
    2        S    0.307544   0.000000
    3        X    0.000000   0.000000
    4        Y    0.000000   0.000000
    5        Z    0.032894   0.000000
    6        S    0.000000   0.763386
    7        S    0.000000   0.307544
    8        X    0.000000   0.000000
    9        Y    0.000000   0.000000
   10        Z    0.000000  -0.032894



                 ******  ORBITAL OVERLAP  ******


            1          2
   1     1.000000   0.712623
   2     0.712623   1.000000



              ******        DENSITY MATRIX         ******


                      1          2          3          4          5

    1        S    0.360753
    2        S    0.145336   0.058551
    3        X    0.000000   0.000000   0.000000
    4        Y    0.000000   0.000000   0.000000   0.000000
    5        Z    0.015545   0.006262   0.000000   0.000000   0.000670
    6        S    0.311533   0.125507   0.000000   0.000000   0.013424
    7        S    0.125507   0.050563   0.000000   0.000000   0.005408
    8        X    0.000000   0.000000   0.000000   0.000000   0.000000
    9        Y    0.000000   0.000000   0.000000   0.000000   0.000000
   10        Z   -0.013424  -0.005408   0.000000   0.000000  -0.000578

                      6          7          8          9         10

    6        S    0.360753
    7        S    0.145336   0.058551
    8        X    0.000000   0.000000   0.000000
    9        Y    0.000000   0.000000   0.000000   0.000000
   10        Z   -0.015545  -0.006262   0.000000   0.000000   0.000670
 ISCF = 5 currently does not support VB orbital densities


                  ===============================================
                          XMVB ATOMIC POPULATION ANALYSIS
                  ===============================================


                      ******  POPULATION AND CHARGE  ******

       ATOM                MULL.POP.    CHARGE          LOW.POP.     CHARGE
    1 H             1.000000    0.000000         1.000000    0.000000
    2 H             1.000000   -0.000000         1.000000   -0.000000

                      ******  ATOMIC SPIN POPULATION ******

     ATOM           MULL.POP.                    LOW.POP.

      1   H           0.000000                     0.000000
      2   H           0.000000                     0.000000


                         ******  BOND ORDER  ******

               ATOM 1        ATOM 2           DIST     BOND ORDER

                   1 H           2 H            0.740       0.952


                      ******    VALENCE ANALYSIS    ******

                  TOTAL       BONDED        FREE
               ATOM            VALENCE     VALENCE     VALENCE
             1 H               1.000       0.952       0.048
             2 H               1.000       0.952       0.048

                      ****** DIPOLE MOMENT ANALYSIS ******

             DX         DY         DZ        TOTAL

           0.000000    0.000000   -0.000000    0.000000


                  ****** ENERGY DECOMPOSITION ANALYSIS ******

                   TOTAL VB ENERGY :          -1.146605431571
               NUCLEAR REP. ENERGY :           0.715104335541
                    KINETIC ENERGY :           1.134053760016
                  POTENTIAL ENERGY :          -2.280659191588
               VARIAL THEOREM VALUE:           2.011067968731



 Cpu for the Job:       0.38 (sec)

 Job Finished at Mon Jun  8 14:36:17 2015

1.3.2. File with optimized VB orbitals (.orb) and orbital guess (.gus)

A file with extension “orb” is an output file of XMVB, which stores the optimized VB orbitals. The format is as follows:

max(1), max(2), . . . , max(val3)
# comment for orbital 1
cvic(1,1), nvic(1,1), cvic(1,1), nvic(2,1), . . . , cvic(max(1),1), nvic(max(1),1)
# comment for orbital 2
cvic(1,2), nvic(1,2), cvic(2,2), nvic(2,2), . . . , cvic(max(2),2), nvic(max(2),2)
. . .
# comment for orbital n
cvic(1,val3), nvic(1,val3), cvic(2,val3), nvic(2, val3), . . . , cvic(max(val3), val3), nvic(max(val3), val3)

where max(i) stands for the number of basis functions in $i^\textrm{th}$ VB orbital, nvic(j,i) is the $j^\textrm{th}$ basis function in $i^\textrm{th}$ VB orbital and cvic(j,i) is the coefficient of nvic(j,i). The lines starting with “#” are treated as comments.

For VBSCF and BOVB calculations, a file of orbital guess may be provided. For VBCI calculations, the guess from a previous VBSCF calculation is required with the extension “gus”. The format of orbital guess file is exactly the same as ORB file. Initial guess files with or without comments are both supported by XMVB.

Note

The initial guess from previous computation with GUESS=READ or GUESS=RDCI now is recommended to be provided in Initial guess description ($GUS) section rather than in an external file to make the computation simpler. The support for GUS file remains mainly for the compatibility.

1.3.3. File with additional information (.xdat)

The file with extension “xdat” is an output file of XMVB. It keeps some other information such as the orbitals in original basis form. Using utility overview:viewing vb orbitals: moldendat can read this file and put the VB orbitals to Gaussian and GAMESS output files and Gaussian fchk files.

1.3.4. One-electron density file (.den)

If keyword DEN of Global control ($CTRL) section, hybrid VB methods (DFVB, VBEFP, VBPCM and VBEFPPCM), or printing level larger than 1 are specified, one-electron density is saved to a file with “den” extension.

1.3.5. File with basis functions information (.info)

This file stores the information of basis functions for the current system, including number of atoms, number of primitive basis functions, basis functions for each atom, and the type of each basis function. This file is essential for stand-alone distribution to carry out FRGTYP=ATOM or FRGTYP=SAO and population analysis.

1.3.6. File with coefficients for the structures/determinants (.coeff)

This file will be obtained after a required TBVBSCF calculation. The coefficients for the structures/determinants are stored in the file and it may be used for later TBVBSCF to accelerate solving secular equation which is proceeded by Davidson Diagnolazation. If the number of structures is larger than the number stored in “coef”, they will be treated as coefficients of the first N structures and the rest will be set to zero.

1.4. Theory and Methodologies

In this appendix, a brief introduction to VB theory and methodologies will be given to the users. For more detailed information, it is recommended to the users to read our reviews and research papers.

1.4.1. Introduction to VB Theory

In quantum chemistry, the many-electron wave function for a system is expressed as a linear combination of state functions:

\[\Psi=\sum_{K}C_K\Phi_K\]

In spin-free quantum chemistry, state functions $\Phi_{K}$ should be a spin eigenfunction with anti-symmetry with respect to permutation of electron indices.The wave function is of the form

\[\Phi_K=\hat{A}\Omega_0\Theta_K\]

where $\hat{A}$ is an antisymmetrizer, $\Omega_0$ is an orbital product as

\[\Omega_0=\phi_1(1)\phi_2(2)\cdots\phi_N(N)\]

where $\phi_i$ is the set of VB orbitals which can be purely localized hybrid atomic orbitals (HAOs), bond distorted orbitals (BDOs, delocalized along the bonding direction), and totally delocalized overlap enhanced orbitals (OEOs), and $\Theta_{K}$ is a spin function. For VB methods, the state functions are VB functions, and their spin functions may be taken as the Rumer basis sets

\[\begin{split}\begin{aligned}\Theta_K&=2^{-1/2}\left[\alpha(i_1)\beta(j_1)-\beta(i_1)\alpha(j_1)\right]\times2^{-1/2}\left[\alpha(i_2)\beta(j_2)-\beta(i_2)\alpha(j_2)\right]\cdots\\&=\prod_{(ij)}2^{-1/2}\left[\alpha(i)\beta(j)-\beta(i)\alpha(j)\right]\prod_k\alpha(k)\end{aligned}\end{split}\]

where $\left(ij\right)$ runs over all bonds and k over all unpaired electrons. Given an orbital product $\Omega_0$ a complete set of VB functions is constructed by choosing all independent spin functions $\Theta_{K}$ .

The coefficients $C_{K}$ in Eq. (A.1) are determined by solving the conventional secular equation $\mathbf{HC}=E\mathbf{MC}$ , where Hamiltonian and overlap matrices are defined as follows:

\[H_{KL}=\langle\Phi_K|H|\Phi_L\rangle\]

and

\[M_{KL}=\langle\Phi_K|\Phi_L\rangle\]

Structural weights are given by the Coulson-Chirgwin formula

\[W_K=\sum_{L}C_KM_{KL}C_L\]

Eqs. (A.5) and (A.6) involve N! terms due to antisymmetrizer $\hat{A}$ . If one-electron functions are orthogonal, only a few terms are non-zero and make contributions to the matrix elements, and consequently the matrix elements can be conveniently evaluated. However, in VB methods, non-orthogonal orbitals are generally used, and thus all N! terms make contributions to the matrix elements. Although it is not necessary to expand all N! terms to evaluate a determinant, the computational demanding in VB calculations is in general much more than that in MO calculations.

1.4.2. The Evaluation of Hamiltonian and Overlap Matrices

In the XMVB package, two algorithms are implemented to compute the Hamiltonian and overlap matrices: one based on the Slater determinant expansion method, and the other based on the paired-permanent-determinant method.

1.4.2.1. Slater determinant expansion algorithm

Traditionally, an HLSP function is expressed in terms of $2^{m}$ Slater determinants (m is the number of covalent bonds of structure),

\[\Phi_K=\prod_i{(1-P_i)}D(\Omega_K)\]

where $D(\Omega_K)$ is a Slater determinant corresponding to Eq. (A.3), $P_{i}$ is an operator that exchanges the spins of the two electrons forming the i-th bond.

Example: An HLSP function corresponding to a Kekulé structure of benzene is written as

\[\begin{gathered}\Phi_{K} =|a\bar{b}c\bar{d}e\bar{f}|-|\bar{a}bc\bar{d}e\bar{f}|-|a\bar{b}\bar{c}de\bar{f}|+|\bar{a}b\bar{c}de\bar{f}|-|a\bar{b}c\bar{d}\bar{e}f|+|\bar{a}bc\bar{d}\bar{e}f|+|a\bar{b}\bar{c}d\bar{e}f|-|\bar{a}b\bar{c}d\bar{e}f|\end{gathered}\]

The Hamiltonian matrix element is expressed as

\[\langle D_i|H|D_j\rangle=\sum_{r,s}f_{rs}D(S_r^s)+\sum_{r\leq u,s\leq t}(g_{rs,ut}-g_{rs,tu})D(S_{ru}^{st})\]

where $f_{rs}$ and $g_{rs,ut}$ are one-electron and two-electron integrals respectively, and $D(S_r^s)$ and $D(S_{ru}^{st})$ are the first and the second order cofactors of the overlap matrix between the two determinants respectively. Cofactors are computed by the Jacobi ratio theorem. The costs are of the order $N^3$ for the first order and $N^4$ for the second order cofactors at most.

1.4.2.2. Paired-permanent-determinant approach

Paired-permanent-determinant (PPD) approach is based on the spin-free form of VB theory. In the spin-free VB theory, the Hamiltonian and overlap matrix elements are now written as

\[H_{KL}=\langle\Phi_K|H|\Phi_L\rangle=\sum_{P\in S_N}D_{11}^{[\lambda]}(P)\langle\Omega_K|HP|\Omega_L\rangle\]

and

\[M_{KL}=\langle\Phi_K|\Phi_L\rangle=\sum_{P\in S_N}D_{11}^{[\lambda]}(P)\langle\Omega_K|P|\Omega_L\rangle\]

respectively, where is the first diagonal element of the standard irreducible representation of permutation P of the symmetric group $S_{N}$. In the PPD approach, a function, called PPD, is defined as follow:

Given an N × N square matrix

\[\begin{aligned}\mathbf{A}=\{a_{ij},i,j=1,2,\cdots,N\}\end{aligned}\]

the PPD of $\mathbf{A}$ for the irreducible representation [λ] is the number

\[\mathrm{ppd}(\lambda,\mathbf{A})=\sum_{P\in S_N}D^{[\lambda]}11(P)a_{1_{p_1}}a_{2_{p_2}}\cdots a_{N_{p_N}}\]

The evaluation of a PPD function is performed by a procedure similar to the Laplacian expansion algorithm for determinant. Hamiltonian and overlap matrix elements are computed by multiplying electronic integrals with their corresponding cofactors of PPDs. Evaluation of a PPD is more complicated than that of a determinant. But it can be beneficial when there are many bonded pairs in system. In that case there are only a few PPDs rather than numerous determinants to be evaluated.

1.4.3. Orbital Optimization

The gradient vectors of energy are evaluated in four ways: the first is the numerical approximation by differential method; the second is analytical gradient based on Fock matrices, using only the first order density matrix; the third is analytical based on the first and the second order orbital density matrices; and the third is based on generalized Brillouin theorem. The first three methods are fitted for all-type orbitals, and the later one is only available for strictly localized and delocalized orbitals. The second one is suitable only when there is no orthogonality between VB functions. There are two orbital optimization methods adopted in the package. The optimization with numerical gradient is based on the Davidson-Fletcher-Powell (DFP) family of variable metric methods, and the optimization with analytical gradient is proceeded with limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method.

1.4.4. The VBSCF Methods

The wave function of Valence Bond Self Consistent Field (VBSCF) method is the linear combination of VB structures, as shown in eq.(A.1). In VBSCF method, All VB structures share the same set of VB orbitals, and both sets of the structure coefficients and VB orbitals are optimized simultaneously to minimize the total energy. This is comparable to the MCSCF method in the MO theory. VBSCF method takes care of the static electron correlation and gives equivalent results to the MO-based CASSCF calculations. It should be noted that the dynamic electron correlation is not accounted for in the VBSCF method. In XMVB, VBSCF method is the default method, thus this keyword can be ignored.

1.4.5. Post-VBSCF Method

The VBSCF result includes only static correlation energy, which makes VBSCF results not accurate enough for quantitative researches. The purpose of post-VBSCF methods is to take dynamic correlation into account as much as possible to get accurate enough results. There are several post-VBSCF methods developed so far and will be introduced in this section. It is strongly recommended to perform post-VBSCF calculations with initial guesses from a pre-proceeded VBSCF calculation. As to VBCI and VBPT2, this is enforced.

1.4.5.1. The BOVB Method

The orbitals of Breathing Orbital Valence Bond (BOVB) method are also optimized by SCF procedure, as VBSCF does. The difference between VBSCF and BOVB methods is that BOVB provides an extra degree of freedom during orbital optimization. In BOVB method, each VB structure has its own set of orbitals and are optimized independently

\[\Psi^{\textrm{VBSCF}} = C_1\left( \vert \phi_a\overline{\phi_b} \vert - \vert \phi_b\overline{\phi_a} \vert \right) + C_2\vert \phi_a\overline{\phi_a} \vert + C_3 \vert \phi_b\overline{\phi_b} \vert\]

\[\Psi^{\textrm{BOVB}} = B_1\left( \vert \phi_a\overline{\phi_b} \vert - \vert \phi_b\overline{\phi_a} \vert \right) + B_2\vert \phi'_a\overline{\phi'_a} \vert + C_3 \vert \phi'_b\overline{\phi'_b} \vert\]

Thus, the orbitals adopt themselves to the instantaneous field of the VB structures, rather than to the mean field of all the structures in VBSCF. This degree of freedom makes the orbitals in BOVB “breathing” in different structures, introduces dynamic correlation, and thereby improves considerably the accuracy of VB computations.

1.4.5.2. The VBCI Method

The VBCI method is based on localized VB orbitals. In this method VB orbitals are divided to several blocks (occupied and virtual orbitals). Excited VB structures are generated by replacing occupied VB orbitals with virtual orbitals that are localized on the same block. The wave function of VBCI is the linear combination of all reference and excited VB structures

\[\Psi^{\textrm{VBCI}} = \sum_K\sum_iC_{Ki}\Phi^i_K\]

where $\Phi^i_K$ is CI structure coming from VBSCF structure K, including reference and excited structures, and the coefficients ${C_{Ki}}$ are obtained by solving the secular equation. The VBCI weight can be given either with eq. (A.7), which gives weights of all CI structures, or in a more compact way as

\[W_K = \sum_iW_{Ki}\]

where $W_K$ is the contracted weights of reference structure K, including weights of all CI structures coming from structure K.

Allowing for different excitations for different electronic shells, currently the VBCI method consists of the following calculation levels:

VBCI(S,S): only single excitations are involved in either active electron or inactive electron. In brief, this is a VBCIS procedure.
VBCI(D,S): the active shell is treated by single and double excitations, whereas the inactive shell by single excitations only. Also included in this level are double excitations which consist of a single excitation from each shell.
VBCI(D,D): single and double excitations are involved for both active and inactive electrons, in short, VBCISD.

1.4.5.3. The VBPT2 Method

Another post-VBSCF method is Valence Bond second-order Perturbation Theory (VBPT2) method. The wave function of VBPT2 can be separated into 2 parts as

\[\Psi^{\textrm{VBPT2}} = \Psi^0 + \Psi^1\]

where VBSCF wave function is taken as the zeroth-order wave function $\Psi^0$, and the first-order part is the combination of singly and doubly excited wave functions

\[\Psi^1 = \sum_{R\in V^{SD}}C^1_R\Phi_R\]

To enhance the efficiency of VBPT2, the virtual orbitals are delocalized and orthogonal to the occupied space, and the excitations include all virtual orbitals. In this manner, the excited structures in VBPT2 don’t belong to any fundamental structure, and the matrix elements can be calculated easily with Coulson-Slater rules.

1.4.5.4. The DFVB Method

Density functional valence bond (DFVB) method is a VB computational method which combines VBSCF and DFT correlation functional. In DFVB method, the wave function, density and static correlation energy is provided by VBSCF method, while the dynamic correlation energy is obtained by DFT correlation functional. The total energy formalism of DFVB is expressed as:

\[E^\textrm{DFVB}\approx E^\textrm{VBSCF}+E_\textrm{C}\left[\rho^\textrm{VB}\right]\]

where $E^\textrm{VBSCF}$ is the VBSCF part, including static correlation energy, and $E_\textrm{C}\left[\rho^\textrm{VB}\right]$ is the DFT correlation energy obtained by a functional according to current VB density $\rho^\textrm{VB}$ . The total energy and wave function is optimized through a SCF procedure. In practice, GGA correlation functionals, such as LYP, PW, PBE correlation functional, are recommended functionals.

1.4.6. Solvation VB Methods

1.4.6.1. The VBPCM Method

The VBPCM method is an ab initio solvation VB method that is based on implicit solvation model PCM in which the state wave function is expressed in the usual terms as a linear combination of VB structures. The Schrödinger equation of VBPCM is expressed as

\[\left(\hat{H}^0+\hat{V}^{\textrm{PCM}}\right)\Psi^{\textrm{VBPCM}}=E^{\textrm{VBPCM}}\Psi^{\textrm{VBPCM}}\]

where $\hat{H}^0$ is the Hamiltonian operator in vacuum and $\hat{V}^{\textrm{PCM}}$ is the solvation potential obtained by PCM. The VBPCM wave function and energy are optimized simultaneously in an SCF procedure. VBPCM is now available for VBSCF and BOVB.

VBPCM has been used in several researches. VBPCM has been rewritten in XMVB 2.0 and is capable for hetero-PCM and EFP/PCM calculations.

1.4.6.2. The VBEFP Method

The VBEFP method is an QM/MM method in which the QM part is expressed as a VB wave function and the MM part is expressed with EFP1, which is a polarized water model proposed by Gordon et al. The VBEFP energy is obtained by following equation

\[\left(\hat{H}^0+\hat{V}^{\textrm{EFP}}\right)\Psi^{\textrm{VBEFP}}=E^{\textrm{VBEFP}}\Psi^{\textrm{VBEFP}}\]

where $\hat{H}^0$ is the Hamiltonian operator in vacuum and $\hat{V}^{\textrm{EFP}}$ is the solvation potential obtained by EFP method. An SCF procedure is used to optimize the wave function and energy of VBEFP simultaneously.

Currently, VBEFP is available only in module distribution and is only available for VBSCF. With the use of EFP, VBEFP is very useful to take strong and short solvent-solute interactions into account.

2. Tutorial

2.1. BDE and RE of F₂

2.1.1. Introduction

F₂ is a typical diatimic molecule which is simple enough for the users as a starting point. There is only 1 chemical bonding in the molecule. The 3 structures of the bonding is shown below.

_images/f2.jpg — Fig. 2.1.1 Structures of F₂ molecule

Here S1 denotes the covalent structure in which two active electrons are shared between both F atoms while S2 and S3 denote 2 quivalent ionic structures in which two active eletrons doubly occupy orbital on certain F atom.

In this exercise, we will try to proceed computations for the bond dissociation energy (BDE) and resnance energy (RE) of F2. This takes computations at stationary point and dissociation limit with various sets of VB structures. This exercise will provide a first glance at VB computation. The users will learn the struct and syntax of XMVB input, how to build a simple input file, how to run the job and what we will get from the output.

Note

This exercise just shows the users how to proceed a VBSCF computation for a specific molecule, what we can get from the output and how to analyze the results. The accurate computation of BDE of F₂ requires higher level computational methods with delocalized inactive $\pi$ orbitals.

2.1.2. Computations with 3 Structures at Stationary Point

The computations are proceeded with F-F bond length 1.4 Angstrom, and the basis set is cc-pVDZ. For simplicity, F atoms are located in the Z axis.

2.1.2.1. Input File

Here shows the XMVB input file for all 3 structures at stationary point:

F2 VBSCF with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 ISCF=5 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
INT=LIBCINT BASIS=CC-PVDZ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*10
1
2
1
2
3
4
5
6
1
2
$END
$GEO
F 0.0 0.0 0.0
F 0.0 0.0 1.4
$END

The global keywords listed in $CTRL section are explained below:

STR=FULL XMVB generates all VB structures automatically according to a specific active space.
ISCF=5 VBSCF computation with RDM-based algorithm.
IPRINT=3 XMVB will print most information.
NAO=2 and NAE=2 Specify the active space with 2 active orbital and 2 active electrons respectively.
ORBTYP=HAO and FRGTYP=SAO The VB orbitals are described with fragments.
INT=LIBCINT Integrals are evaluated by XMVB with LIBCINT library.
BASIS=CC-PVDZ The basis set is cc-pVDZ.

Note

The orbitals of a molecule can be devided into “inactive” and “active” parts. The inactive orbitals are always doubly occupied in all VB structures, while the occupation numbers of active orbitals can be 0, 1 or 2 in each VB structure.

$FRAG section describes the fragments used to construct VB orbitals. In this case, the first line 1*6 means that there are 6 fragments built and each fragment include only 1 atom. The details of fragments are described in the following lines. For example, SPZDXXDYYDZZ 1 means that the fragment includes s, p_z, d_xx, d_yy and d_zz basis functions on atom 1, which is the first F atom in this case.

$ORB section describes VB orbitals. The first line 1*10 means that there are 10 orbitals in this case, each consisting of only 1 fragment. Each line below describes one orbital. For example, the first orbital includes only fragment 1, meaning that this orbital locates on the first F atom and belongs to the $\sigma$ space since the F-F lies on Z axis. Also, orbital 6 with fragment 4 shows that the orbital describes the $\pi_x$ orbital on the first F atom.

$GEO section shows the geometry of F₂ molecule. Both Cartesian and internal coordinates are supported.

Note

Since NAO=2 is specified in $CTRL section, the last 2 orbitals in $ORB should be the active orbitals.

It is highly recommended that active orbitals are always palced after the inactive ones. This may make the input file more readable and less possible to get error.

2.1.2.2. Computational Results

In this case, 3 VB structures are generated with 1 covalent structure (0th ion structure) and 2 ionic structures (1th ion structures). These structures can be expaned into 4 determinants.

Number of   0th ion  structures  is:          1  from          1 to          1
Number of   0th ion determinants is:          2  from          1 to          2
Number of   1th ion  structures  is:          2  from          2 to          3
Number of   1th ion determinants is:          2  from          3 to          4
Total number of  structures  is:          3
Total number of determinants is:          4
                   .
                   .
                   .
Number of Structures:           3

The following structures are used in calculation:

      1 *****  1:8    9  10
      2 *****  1:8    9   9
      3 *****  1:8   10  10

The user may find from the output file that the VBSCF converged after 25 iterations and the final energy is -198.75115493 hartree.

                       VBSCF converged in    25 iterations

Total Energy:    -198.75115493

First Excited:   -197.880879

The coefficients and weights shows the importance of each structure. Following are the coefficients and weights of generated VB structures. Both coefficients and weights show that the covalent structure is the dominant one. So the F-F bond should be a covalent bond.

       ******  COEFFICIENTS OF STRUCTURES ******

1    -0.80579  ******  1:8    9  10
2    -0.21326  ******  1:8    9   9
3    -0.21326  ******  1:8   10  10
                  .
                  .
                  .
       ******  WEIGHTS OF STRUCTURES ******

1     0.77586  ******  1:8    9  10
2     0.11207  ******  1:8    9   9
3     0.11207  ******  1:8   10  10

Finally, the bond order value (0.773) in atomic population analysis shows that F-F bond should be a single bond.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 F           2 F            1.400       0.773

2.1.3. Computations with 3 Structures at Dissociation Limit

2.1.3.1. Input File

The input file for such computation can be easily obtained by modifying the corresponding one at stationary point. As shown below, one just needs to modify the coordinate of the second F atom to change 1.4 to 10.0. This will increase the F-F bond distance to 10.0 Angstrom.

$GEO
F 0.0 0.0 0.0
F 0.0 0.0 10.0
$END

2.1.3.2. Computational Results

After SCF procedure, the user will get the final energy as shown below. The computational energy of dissociated F₂ is -198.74386524 hartree.

Total Energy:    -198.74386524

First Excited:   -197.850852

Both coefficients and weights of VB structures show that the wave function includes only covalent structure, indicating that the molecule is dissociated to 2 F$\cdot$ radicals.

       ******  COEFFICIENTS OF STRUCTURES ******

1     1.00000  ******  1:8    9  10
2     0.00000  ******  1:8    9   9
3     0.00000  ******  1:8   10  10
                      .
                      .
                      .
       ******  WEIGHTS OF STRUCTURES ******

1     1.00000  ******  1:8    9  10
2     0.00000  ******  1:8    9   9
3     0.00000  ******  1:8   10  10

By substracting energies at stationary point and dissociation limit, the BDE can be obtained as 4.4 kcal/mol.

2.1.4. Computations with Covalent Structure at Stationary Point

2.1.4.1. Input File

The input file can be obtained by replacing STR=FULL with STR=COV in the input file of 3 structures at stationary point, as shown below.

$CTRL
STR=COV NAO=2 NAE=2 ISCF=5 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
INT=LIBCINT BASIS=CC-PVDZ
$END

2.1.4.2. Computational Results

The structure information is shown below. It can be seen that only 1 covalent structure is generated for this computation.

Number of   0th ion  structures  is:          1  from          1 to          1
Number of   0th ion determinants is:          2  from          1 to          2
Total number of  structures  is:          1
Total number of determinants is:          2
                       .
                       .
                       .
 Number of Structures:           1

 The following structures are used in calculation:

       1 *****  1:8    9  10

The final energy after SCF iteration is -198.67485969 hartree.

Total Energy:    -198.67485969

The Last Change in Energy:  -0.000000

The bond order result in atomic population analysis shows that the bond order with only covalent structure is 0.324, which is much smaller than computations with 3 structures.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 F           2 F            1.400       0.324

The RE value, 47.6 kcal/mol, can be obtained from energies by covalent and 3 structure computations at stationary point. Compared with the computational BDE 4.4, RE is much larger than BDE, indicating that the chemical bonding in F₂ is an charge-shifting bond. The much smaller bond order data with only covalent structure also indicates that resonance plays an important role in the F-F bonding.

2.2. Resonance in C₆H₆

2.2.1. Introduction

Benzene (C₆H₆) is a classical aromatic molecule in which electrons in $\pi$ space are delocalized. The covalent valence bond structures are shown in the following table.

_images/c6h6.eps.png — Fig. 2.2.1 Covalent Structures of C₆H₆

The 5 covalent structures can be classified as 2 types: the first 2 structures are Kekulé structures and the others are regarded as Dewar structures. In this exercise, the resonance energy in the benzene molecule will be computed to show how resonance stabilize the aromatic molecule.

2.2.2. Computations with 5 Covalent Structures

First, we will compute the energy with all 5 covalent structures.

2.2.2.1. Input File

The input file is shown below.

C6H6
$ctrl
str=cov nao=6 nae=6 iscf=5 iprint=3
orbtyp=hao frgtyp=sao
int=libcint basis=cc-pvdz
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
 C           0.6995000584   1.2115696411   0.0000000000
 H           1.2460106991   2.1581538376   0.0000000000
 C          -0.6995000584   1.2115696411   0.0000000000
 H          -1.2460106991   2.1581538376   0.0000000000
 C          -1.3990001169   0.0000000000   0.0000000000
 H          -2.4920213982   0.0000000000   0.0000000000
 C          -0.6995000584  -1.2115696411   0.0000000000
 H          -1.2460106991  -2.1581538376   0.0000000000
 C           0.6995000584  -1.2115696411   0.0000000000
 H           1.2460106991  -2.1581538376   0.0000000000
 C           1.3990001169   0.0000000000   0.0000000000
 H           2.4920213982   0.0000000000   0.0000000000
$end

In this input file, the molecule lays in the XY plane. The fragments specified in $FRAG section are defined according to the symmetries. Since we concern on the resonance in $\pi$space, electrons locating in $\sigma$space is not in the interest. Thus eletrons in the $\sigma$orbitals can be treated as inactive ones and make them delocalized in all structures. To this reason, the first fragment is defined to include all basis functions in $\sigma$ space in the molecule. The remained fragments denote $\pi$ orbitals locates on each CH group.

2.2.2.2. Computational Results

5 VB structures will be generated in the computation and can be found in the output:

Number of Structures:           5

The following structures are used in calculation:

      1 *****  1:18  21  22  20  23  19  24
      2 *****  1:18  20  21  22  23  19  24
      3 *****  1:18  20  21  19  22  23  24
      4 *****  1:18  19  20  22  23  21  24
      5 *****  1:18  19  20  21  22  23  24

Final energy can be found after the VBSCF iteration as -230.63450 hartree.

Total Energy:    -230.63449533

Both coefficients and weights of VB structures show that structures are in 2 groups. Structure 2 and 5 correspond to the Kekulé structures, which are the dominant ones, and the others denote Dewar structures.

       ******  COEFFICIENTS OF STRUCTURES ******

1     0.14537  ******  1:18  21  22  20  23  19  24
2    -0.40098  ******  1:18  20  21  22  23  19  24
3     0.14537  ******  1:18  20  21  19  22  23  24
4     0.14537  ******  1:18  19  20  22  23  21  24
5    -0.40098  ******  1:18  19  20  21  22  23  24

       ******  WEIGHTS OF STRUCTURES ******

1     0.11245  ******  1:18  21  22  20  23  19  24
2     0.33132  ******  1:18  20  21  22  23  19  24
3     0.11245  ******  1:18  20  21  19  22  23  24
4     0.11245  ******  1:18  19  20  22  23  21  24
5     0.33132  ******  1:18  19  20  21  22  23  24

The atomic population analysis shows that the bond orders of all CC bondings are the same as 1.188, showing the effect of conjugation.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 C           2 H            1.093       0.979
 1 C           3 C            1.399       1.188
 3 C           4 H            1.093       0.979
 3 C           5 C            1.399       1.188
 5 C           6 H            1.093       0.979
 5 C           7 C            1.399       1.188
 7 C           8 H            1.093       0.979
 7 C           9 C            1.399       1.188
 9 C          10 H            1.093       0.979
 1 C          11 C            1.399       1.188
 9 C          11 C            1.399       1.188
11 C          12 H            1.093       0.979

2.2.3. Computations with 2 Kekulé Structures

2.2.3.1. Input File

The input file is shown below.

C6H6
$ctrl
nstr=2 nao=6 nae=6 iscf=5 iprint=3
orbtyp=hao frgtyp=sao
int=libcint basis=cc-pvdz
guess=read
$end
$str
1:18 19-24
1:18 19 24 20 21 22 23
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
 C           0.6995000584   1.2115696411   0.0000000000
 H           1.2460106991   2.1581538376   0.0000000000
 C          -0.6995000584   1.2115696411   0.0000000000
 H          -1.2460106991   2.1581538376   0.0000000000
 C          -1.3990001169   0.0000000000   0.0000000000
 H          -2.4920213982   0.0000000000   0.0000000000
 C          -0.6995000584  -1.2115696411   0.0000000000
 H          -1.2460106991  -2.1581538376   0.0000000000
 C           0.6995000584  -1.2115696411   0.0000000000
 H           1.2460106991  -2.1581538376   0.0000000000
 C           1.3990001169   0.0000000000   0.0000000000
 H           2.4920213982   0.0000000000   0.0000000000
$end
$gus
    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90     5     5     5     5     5     5
# ORBITAL          1  NAO =      90
-0.4085807072     1  -0.0018715270     2   0.0001649808     3   0.0000999746     4
 0.0001985368     5   0.0000470242     7   0.0000880828     8   0.0009574571    10
 0.0000833088    11   0.0010378240    13   0.0011389938    15  -0.0001957425    16
-0.0001334195    17   0.0000477351    18   0.0000812855    19  -0.4085807076    21
-0.0018715190    22   0.0001649828    23  -0.0000999676    24   0.0001985300    25
-0.0000470232    27   0.0000880811    28   0.0009574481    30  -0.0000833266    31
 0.0010378340    33   0.0011390026    35  -0.0001957414    36  -0.0001334204    37
-0.0000477371    38   0.0000812776    39  -0.4090845359    41  -0.0018966521    42
 0.0001488619    43  -0.0002148556    44   0.0000001381    45  -0.0000988751    47
 0.0000000169    48   0.0010810968    50  -0.0000001656    51   0.0009153322    53
 0.0011399821    55  -0.0002143575    56  -0.0001372115    57  -0.0000966788    58
 0.0000001529    59  -0.4085807085    61  -0.0018714717    62   0.0001649779    63
-0.0001000242    64  -0.0001983775    65  -0.0000470512    67  -0.0000879863    68
 0.0009574723    70   0.0000839012    71   0.0010378364    73   0.0011390336    75
-0.0001956328    76  -0.0001334448    77  -0.0000479296    78  -0.0000811013    79
-0.4085807082    81  -0.0018714747    82   0.0001649755    83   0.0001000278    84
-0.0001983820    85   0.0000470479    87  -0.0000879885    88   0.0009574749    90
-0.0000838796    91   0.0010378346    93   0.0011390280    95  -0.0001956337    96
-0.0001334457    97   0.0000479178    98  -0.0000811144    99  -0.4090845352   101
-0.0018966413   102   0.0001488609   103   0.0002148576   104   0.0000001396   105
 0.0000988788   107   0.0000000209   108   0.0010810705   110   0.0000001777   111
 0.0009153543   113   0.0011399863   115  -0.0002143672   116  -0.0001372134   117
 0.0000966809   118   0.0000001629   119
# ORBITAL          2  NAO =      90
 0.2888930861     1   0.0012304770     2  -0.0002563966     3   0.0001336859     4
-0.0001561191     5   0.0001914831     7  -0.0001481897     8  -0.0008221343    10
-0.0001557600    11  -0.0005906595    13  -0.0007980351    15   0.0001543642    16
 0.0000791296    17   0.0000125324    18  -0.0000788689    19  -0.2888931497    21
-0.0012304753    22   0.0002562013    23   0.0001337942    24   0.0001566737    25
 0.0001915140    27   0.0001494777    28   0.0008219804    30  -0.0001516843    31
 0.0005907316    33   0.0007979421    35  -0.0001550608    36  -0.0000790540    37
 0.0000136909    38   0.0000800072    39  -0.5781904466    41  -0.0024581181    42
 0.0005050109    43  -0.0001390292    44   0.0000000205    45  -0.0000746810    47
 0.0000000140    48   0.0015287631    50  -0.0000000553    51   0.0012982685    53
 0.0015971801    55  -0.0003048222    56  -0.0001585226    57  -0.0001238936    58
 0.0000000479    59  -0.2888930865    61  -0.0012304541    62   0.0002561993    63
 0.0001337970    64  -0.0001566499    65   0.0001915245    67  -0.0001494674    68
 0.0008219645    70   0.0001517465    71   0.0005907569    73   0.0007979557    75
-0.0001550484    76  -0.0000790645    77   0.0000136980    78  -0.0000799949    79
 0.2888931500    81   0.0012304673    82  -0.0002563957    83   0.0001337007    84
 0.0001561303    85   0.0001914997    87   0.0001482011    88  -0.0008221145    90
 0.0001556889    91  -0.0005906834    93  -0.0007980414    95   0.0001543768    96
 0.0000791346    97   0.0000125801    98   0.0000788956    99   0.5781904449   101
 0.0024591332   102  -0.0005052562   103  -0.0001393968   104   0.0000000109   105
-0.0000739491   107  -0.0000000104   108  -0.0015297884   110  -0.0000000183   111
-0.0012968826   113  -0.0015966043   115   0.0003041670   116   0.0001581403   117
-0.0001241023   118  -0.0000000148   119
# ORBITAL          3  NAO =      90
 0.5006118878     1   0.0021171792     2  -0.0004445363     3  -0.0001718795     4
-0.0000528411     5  -0.0001543791     7   0.0000142568     8  -0.0010905424    10
-0.0000444470    11  -0.0013582539    13  -0.0013824483    15   0.0002505625    16
 0.0001341232    17  -0.0000778672    18  -0.0000745639    19   0.5006118515    21
 0.0021171388    22  -0.0004445339    23   0.0001718621    24  -0.0000528197    25
 0.0001543481    27   0.0000142554    28  -0.0010905019    30   0.0000444391    31
-0.0013583104    33  -0.0013824752    35   0.0002505629    36   0.0001341408    37
 0.0000778110    38  -0.0000745770    39  -0.0000000362    41   0.0000000221    42
-0.0000000048    43   0.0000000087    44   0.0002083308    45  -0.0000000157    47
 0.0002704667    48  -0.0000000212    50   0.0002206198    51   0.0000000294    53
 0.0000000132    55  -0.0000000141    56  -0.0000000081    57   0.0000000045    58
 0.0000602462    59  -0.5006118875    61  -0.0021171392    62   0.0004445257    63
-0.0001718574    64  -0.0000528443    65  -0.0001543471    67   0.0000142010    68
 0.0010904959    70   0.0000442655    71   0.0013583140    73   0.0013824713    75
-0.0002505931    76  -0.0001341382    77  -0.0000777637    78  -0.0000746227    79
-0.5006118506    81  -0.0021171793    82   0.0004445278    83   0.0001718746    84
-0.0000528666    85   0.0001543776    87   0.0000142018    88   0.0010905358    90
-0.0000442668    91   0.0013582572    93   0.0013824439    95  -0.0002505929    96
-0.0001341203    97   0.0000778157    98  -0.0000746149    99   0.0000000372   101
 0.0000000233   102  -0.0000000051   103  -0.0000000081   104   0.0002082947   105
 0.0000000157   107   0.0002704090   108  -0.0000000235   110  -0.0002208132   111
 0.0000000315   113   0.0000000132   115  -0.0000000150   116  -0.0000000085   117
-0.0000000048   118   0.0000600832   119
# ORBITAL          4  NAO =      90
-0.2891567875     1  -0.0011788380     2   0.0006696401     3   0.0001162136     4
-0.0000434029     5   0.0000569096     7  -0.0002181218     8   0.0005581052    10
-0.0000173221    11   0.0007001594    13   0.0007639834    15  -0.0000761592    16
-0.0000306619    17   0.0000222375    18   0.0000003327    19  -0.2891568478    21
-0.0011788552    22   0.0006696375    23  -0.0001162265    24  -0.0000433901    25
-0.0000569145    27  -0.0002181183    28   0.0005581234    30   0.0000173582    31
 0.0007001375    33   0.0007639670    35  -0.0000761610    36  -0.0000306572    37
-0.0000222376    38   0.0000003522    39   0.5781677329    41   0.0024208733    42
-0.0012969300    43  -0.0000049661    44  -0.0000000187    45  -0.0003380963    47
 0.0000000028    48  -0.0013034129    50   0.0000000076    51  -0.0012070190    53
-0.0015287103    55   0.0002187599    56   0.0000735925    57   0.0000329236    58
-0.0000000083    59  -0.2891567879    61  -0.0011788549    62   0.0006696380    63
-0.0001162154    64   0.0000433692    65  -0.0000569033    67   0.0002181026    68
 0.0005581117    70  -0.0000174532    71   0.0007001480    73   0.0007639674    75
-0.0000761793    76  -0.0000306569    77  -0.0000221958    78  -0.0000003814    79
-0.2891568468    81  -0.0011788535    82   0.0006696419    83   0.0001162088    84
 0.0000433741    85   0.0000569103    87   0.0002181062    88   0.0005581088    90
 0.0000174178    91   0.0007001487    93   0.0007639740    95  -0.0000761773    96
-0.0000306542    97   0.0000222162    98  -0.0000003606    99   0.5781677327   101
 0.0024208558   102  -0.0012969285   103   0.0000049612   104  -0.0000000253   105
 0.0003380902   107  -0.0000000083   108  -0.0013033682   110  -0.0000000436   111
-0.0012070582   113  -0.0015287170   115   0.0002187773   116   0.0000735959   117
-0.0000329255   118  -0.0000000388   119
# ORBITAL          5  NAO =      90
-0.5007497195     1  -0.0020926701     2   0.0011274729     3  -0.0000664898     4
 0.0000686296     5  -0.0002263515     7  -0.0001918806     8   0.0011015128    10
 0.0000674501    11   0.0010736729    13   0.0013237329    15  -0.0001590050    16
-0.0000498765    17   0.0000054571    18   0.0000298585    19   0.5007496847    21
 0.0020926790    22  -0.0011275085    23  -0.0000664796    24  -0.0000685616    25
-0.0002263533    27   0.0001920864    28  -0.0011015340    30   0.0000680448    31
-0.0010736610    33  -0.0013237436    35   0.0001589047    36   0.0000498846    37
 0.0000056242    38  -0.0000296943    39   0.0000000348    41   0.0000000890    42
-0.0000000212    43   0.0000000320    44   0.0001851945    45  -0.0000000646    47
 0.0002002088    48  -0.0000000915    50   0.0000942443    51   0.0000001220    53
 0.0000000504    55  -0.0000000578    56  -0.0000000328    57   0.0000000176    58
 0.0000247770    59  -0.5007497199    61  -0.0020926786    62   0.0011274738    63
 0.0000664970    64  -0.0000686556    65   0.0002263560    67   0.0001918646    68
 0.0011015108    70   0.0000673487    71   0.0010736709    73   0.0013237272    75
-0.0001590238    76  -0.0000498723    77  -0.0000054290    78  -0.0000298875    79
 0.5007496855    81   0.0020926723    82  -0.0011275078    83   0.0000664707    84
 0.0000685365    85   0.0002263476    87  -0.0001921026    88  -0.0011015386    90
 0.0000681438    91  -0.0010736605    93  -0.0013237484    95   0.0001588863    96
 0.0000498882    97  -0.0000056552    98   0.0000296659    99   0.0000000336   101
 0.0000000883   102  -0.0000000217   103  -0.0000000323   104  -0.0001852396   105
 0.0000000644   107  -0.0002002133   108  -0.0000000900   110   0.0000941944   111
 0.0000001203   113   0.0000000496   115  -0.0000000565   116  -0.0000000328   117
-0.0000000169   118  -0.0000248210   119
# ORBITAL          6  NAO =      90
 0.4091251475     1   0.0018241997     2  -0.0036592202     3   0.0000020694     4
-0.0000011859     5   0.0007741716     7   0.0013387077     8  -0.0007515405    10
 0.0000812994    11  -0.0006732113    13  -0.0009930082    15  -0.0000359553    16
-0.0002099914    17   0.0000257544    18   0.0000416529    19  -0.4091251464    21
-0.0018242229    22   0.0036594107    23   0.0000019865    24   0.0000007218    25
 0.0007741606    27  -0.0013398977    28   0.0007516717    30   0.0000776616    31
 0.0006731473    33   0.0009930840    35   0.0000365738    36   0.0002099303    37
 0.0000247332    38  -0.0000426613    39   0.4088400900    41   0.0018185067    42
-0.0036610579    43  -0.0000000395    44   0.0000000662    45  -0.0015467853    47
 0.0000000329    48  -0.0006312514    50  -0.0000001499    51  -0.0007924283    53
-0.0009926163    55  -0.0000380211    56  -0.0002097659    57  -0.0000506940    58
 0.0000001297    59  -0.4091251455    61  -0.0018241642    62   0.0036594070    63
 0.0000019879    64  -0.0000006481    65   0.0007741792    67   0.0013399307    68
 0.0007516425    70  -0.0000774650    71   0.0006732027    73   0.0009931240    75
 0.0000366093    76   0.0002099034    77   0.0000247197    78   0.0000427155    79
 0.4091251465    81   0.0018241721    82  -0.0036592195    83   0.0000021021    84
 0.0000012118    85   0.0007742169    87  -0.0013386734    88  -0.0007514874    90
-0.0000814990    91  -0.0006732734    93  -0.0009930272    95  -0.0000359181    96
-0.0002099801    97   0.0000258758    98  -0.0000415851    99  -0.4088400890   101
-0.0018194794   102   0.0036612935   103   0.0000003129   104   0.0000000252   105
-0.0015474894   107  -0.0000000271   108   0.0006322381   110  -0.0000000559   111
 0.0007910987   113   0.0009920653   115   0.0000386489   116   0.0002101309   117
-0.0000504971   118  -0.0000000437   119
# ORBITAL          7  NAO =      90
-0.0107819418     1   0.1790782666     2   0.0924284475     3  -0.0318865151     4
-0.0553049355     5  -0.0062977525     7  -0.0109401070     8   0.0108331095    10
-0.0038054399    11   0.0070287849    13  -0.0063313721    15   0.0455977157    16
 0.0036387887    17  -0.0036035054    18  -0.0062459698    19  -0.0107819418    21
 0.1790782665    22   0.0924284475    23   0.0318865153    24  -0.0553049354    25
 0.0062977526    27  -0.0109401069    28   0.0108331095    30   0.0038054400    31
 0.0070287847    33  -0.0063313720    35   0.0455977157    36   0.0036387887    37
 0.0036035054    38  -0.0062459697    39  -0.0107755990    41   0.1791237848    42
 0.0924406313    43   0.0638465961    44  -0.0000000002    45   0.0126243430    47
 0.0000000000    48   0.0051202404    50   0.0000000002    51   0.0127397005    53
-0.0063317099    55   0.0456056414    56   0.0036342774    57   0.0072116661    58
-0.0000000001    59  -0.0107819418    61   0.1790782665    62   0.0924284475    63
 0.0318865153    64   0.0553049352    65   0.0062977526    67   0.0109401069    68
 0.0108331095    70  -0.0038054405    71   0.0070287846    73  -0.0063313721    75
 0.0455977156    76   0.0036387886    77   0.0036035055    78   0.0062459696    79
-0.0107819418    81   0.1790782665    82   0.0924284475    83  -0.0318865153    84
 0.0553049354    85  -0.0062977525    87   0.0109401069    88   0.0108331095    90
 0.0038054404    91   0.0070287849    93  -0.0063313722    95   0.0455977156    96
 0.0036387887    97  -0.0036035055    98   0.0062459696    99  -0.0107755990   101
 0.1791237842   102   0.0924406314   103  -0.0638465970   104  -0.0000000002   105
-0.0126243426   107   0.0000000000   108   0.0051202413   110  -0.0000000001   111
 0.0127397007   113  -0.0063317099   115   0.0456056422   116   0.0036342781   117
-0.0072116650   118  -0.0000000001   119
# ORBITAL          8  NAO =      90
 0.0080809826     1  -0.1247496483     2  -0.0702559576     3  -0.0784948406     4
 0.0639689094     5  -0.0281707685     7   0.0270146479     8   0.0056247890    10
 0.0119191820    11  -0.0144007347    13   0.0039111625    15  -0.0517266720    16
-0.0069585491    17   0.0014215117    18   0.0074905042    19  -0.0080809832    21
 0.1247496564    22   0.0702559624    23  -0.0784948380    24  -0.0639689094    25
-0.0281707673    27  -0.0270146491    28  -0.0056247882    30   0.0119191787    31
 0.0144007344    33  -0.0039111628    35   0.0517266760    36   0.0069585494    37
 0.0014215110    38  -0.0074905056    39  -0.0161618860    41   0.2495029330    42
 0.1404967375    43   0.0323045190    44   0.0000000022    45   0.0186150565    47
 0.0000000009    48   0.0009097151    50   0.0000000004    51   0.0166398753    53
-0.0078227360    55   0.1034460314    56   0.0139150017    57   0.0143938231    58
 0.0000000000    59  -0.0080809826    61   0.1247496481    62   0.0702559577    63
-0.0784948405    64   0.0639689095    65  -0.0281707683    67   0.0270146490    68
-0.0056247889    70  -0.0119191785    71   0.0144007345    73  -0.0039111625    75
 0.0517266726    76   0.0069585489    77   0.0014215107    78   0.0074905052    79
 0.0080809832    81  -0.1247496565    82  -0.0702559623    83  -0.0784948381    84
-0.0639689091    85  -0.0281707675    87  -0.0270146480    88   0.0056247883    90
-0.0119191823    91  -0.0144007346    93   0.0039111629    95  -0.0517266754    96
-0.0069585496    97   0.0014215120    98  -0.0074905046    99   0.0161618860   101
-0.2495029327   102  -0.1404967374   103   0.0323045205   104   0.0000000024   105
 0.0186150551   107   0.0000000008   108  -0.0009097149   110  -0.0000000002   111
-0.0166398772   113   0.0078227354   115  -0.1034460320   116  -0.0139150025   117
 0.0143938213   118   0.0000000001   119
# ORBITAL          9  NAO =      90
 0.0139974362     1  -0.2160783634     2  -0.1216786927     3   0.0639676065     4
-0.0045967336     5   0.0270145605     7   0.0030327889     8  -0.0179215924    10
 0.0022068301    11   0.0027201810    13   0.0067737890    15  -0.0895686954    16
-0.0120431588    17   0.0074907427    18   0.0100672464    19   0.0139974359    21
-0.2160783579    22  -0.1216786897    23  -0.0639676097    24  -0.0045967367    25
-0.0270145618    27   0.0030327877    28  -0.0179215926    30  -0.0022068299    31
 0.0027201819    33   0.0067737894    35  -0.0895686936    36  -0.0120431585    37
-0.0074907427    38   0.0100672457    39  -0.0000000003    41   0.0000000048    42
 0.0000000027    43   0.0000000006    44  -0.1154605477    45   0.0000000003    47
-0.0437801763    48   0.0000000000    50  -0.0184419859    51   0.0000000003    53
-0.0000000002    55   0.0000000020    56   0.0000000003    57   0.0000000003    58
-0.0029035178    59  -0.0139974362    61   0.2160783627    62   0.1216786924    63
 0.0639676067    64  -0.0045967344    65   0.0270145607    67   0.0030327888    68
 0.0179215924    70  -0.0022068304    71  -0.0027201814    73  -0.0067737896    75
 0.0895686955    76   0.0120431587    77   0.0074907427    78   0.0100672461    79
-0.0139974359    81   0.2160783587    82   0.1216786899    83  -0.0639676095    84
-0.0045967361    85  -0.0270145616    87   0.0030327879    88   0.0179215927    90
 0.0022068297    91  -0.0027201816    93  -0.0067737889    95   0.0895686934    96
 0.0120431584    97  -0.0074907426    98   0.0100672461    99   0.0000000003   101
-0.0000000048   102  -0.0000000027   103   0.0000000006   104  -0.1154605457   105
 0.0000000004   107  -0.0437801767   108  -0.0000000001   110   0.0184419877   111
-0.0000000003   113   0.0000000001   115  -0.0000000020   116  -0.0000000003   117
 0.0000000003   118  -0.0029035175   119
# ORBITAL         10  NAO =      90
-0.0051440589     1   0.1005718952     2   0.0845638529     3  -0.1397337895     4
 0.1141834782     5  -0.0457303867     7   0.0234939964     8   0.0117656669    10
 0.0056709072    11  -0.0100493795    13  -0.0019026433    15   0.0770087430    16
 0.0164532289    17  -0.0072726819    18  -0.0060307741    19  -0.0051440587    21
 0.1005718915    22   0.0845638499    23   0.1397337914    24   0.1141834780    25
 0.0457303869    27   0.0234939969    28   0.0117656673    30  -0.0056709071    31
-0.0100493799    33  -0.0019026433    35   0.0770087402    36   0.0164532285    37
 0.0072726816    38  -0.0060307738    39   0.0102806964    41  -0.2012853772    42
-0.1692333678    43   0.0582639973    44  -0.0000000022    45  -0.0049551221    47
-0.0000000006    48   0.0006914310    50  -0.0000000001    51  -0.0041279010    53
 0.0038040175    55  -0.1542612985    56  -0.0329629875    57  -0.0177440549    58
-0.0000000001    59  -0.0051440589    61   0.1005718952    62   0.0845638530    63
 0.1397337890    64  -0.1141834780    65   0.0457303863    67  -0.0234939964    68
 0.0117656672    70   0.0056709069    71  -0.0100493798    73  -0.0019026434    75
 0.0770087429    76   0.0164532290    77   0.0072726817    78   0.0060307742    79
-0.0051440587    81   0.1005718916    82   0.0845638498    83  -0.1397337917    84
-0.1141834783    85  -0.0457303871    87  -0.0234939968    88   0.0117656669    90
-0.0056709073    91  -0.0100493797    93  -0.0019026432    95   0.0770087402    96
 0.0164532283    97  -0.0072726818    98   0.0060307738    99   0.0102806964   101
-0.2012853761   102  -0.1692333681   103  -0.0582639958   104   0.0000000022   105
 0.0049551210   107   0.0000000006   108   0.0006914310   110  -0.0000000001   111
-0.0041279019   113   0.0038040170   115  -0.1542612996   116  -0.0329629888   117
 0.0177440529   118   0.0000000001   119
# ORBITAL         11  NAO =      90
-0.0088998256     1   0.1742799331     2   0.1465089246     3   0.1142486920     4
-0.0078131644     5   0.0235313039     7  -0.0185720459     8  -0.0034227095    10
-0.0060473270    11   0.0064037031    13  -0.0032944140    15   0.1334935781    16
 0.0285120656    17  -0.0060359574    18  -0.0142465448    19   0.0088998257    21
-0.1742799344    22  -0.1465089259    23   0.1142486881    24   0.0078131605    25
 0.0235313028    27   0.0185720448    28   0.0034227096    30  -0.0060473293    31
-0.0064037026    33   0.0032944147    35  -0.1334935795    36  -0.0285120659    37
-0.0060359581    38   0.0142465441    39  -0.0000000001    41   0.0000000021    42
 0.0000000018    43  -0.0000000007    44  -0.2057016988    45   0.0000000001    47
-0.0593097208    48  -0.0000000000    50  -0.0158714988    51   0.0000000000    53
-0.0000000001    55   0.0000000016    56   0.0000000003    57   0.0000000002    58
-0.0037832706    59  -0.0088998256    61   0.1742799323    62   0.1465089241    63
-0.1142486911    64   0.0078131629    65  -0.0235313036    67   0.0185720454    68
-0.0034227098    70  -0.0060473291    71   0.0064037028    73  -0.0032944147    75
 0.1334935779    76   0.0285120654    77   0.0060359579    78   0.0142465442    79
 0.0088998257    81  -0.1742799352    82  -0.1465089263    83  -0.1142486891    84
-0.0078131618    85  -0.0235313029    87  -0.0185720453    88   0.0034227093    90
-0.0060473274    91  -0.0064037028    93   0.0032944141    95  -0.1334935796    96
-0.0285120659    97   0.0060359576    98  -0.0142465449    99  -0.0000000001   101
 0.0000000020   102   0.0000000018   103   0.0000000006   104   0.2057016950   105
-0.0000000001   107   0.0593097217   108   0.0000000001   110  -0.0158715011   111
-0.0000000001   113  -0.0000000000   115   0.0000000017   116   0.0000000004   117
-0.0000000002   118   0.0037832703   119
# ORBITAL         12  NAO =      90
-0.0037231920     1   0.0111150627     2   0.0324659518     3   0.0947657341     4
 0.1639019892     5   0.0324226748     7   0.0560650241     8   0.0004508783    10
-0.0049970547    11  -0.0045275601    13   0.0010122333    15   0.1490787651    16
 0.0353865179    17  -0.0067083649    18  -0.0116265648    19  -0.0037231920    21
 0.0111150626    22   0.0324659517    23  -0.0947657340    24   0.1639019892    25
-0.0324226745    27   0.0560650241    28   0.0004508780    30   0.0049970544    31
-0.0045275597    33   0.0010122332    35   0.1490787652    36   0.0353865178    37
 0.0067083650    38  -0.0116265647    39  -0.0037179973    41   0.0112563382    42
 0.0325136599    43  -0.1892984480    44  -0.0000000001    45  -0.0647516749    47
-0.0000000000    48  -0.0070415813    50   0.0000000001    51   0.0029723742    53
 0.0010156547    55   0.1491231283    56   0.0353714222    57   0.0134368381    58
-0.0000000001    59  -0.0037231920    61   0.0111150627    62   0.0324659517    63
-0.0947657339    64  -0.1639019894    65  -0.0324226745    67  -0.0560650241    68
 0.0004508780    70  -0.0049970546    71  -0.0045275596    73   0.0010122332    75
 0.1490787651    76   0.0353865177    77   0.0067083651    78   0.0116265647    79
-0.0037231920    81   0.0111150627    82   0.0324659518    83   0.0947657342    84
-0.1639019894    85   0.0324226748    87  -0.0560650241    88   0.0004508784    90
 0.0049970549    91  -0.0045275602    93   0.0010122333    95   0.1490787649    96
 0.0353865178    97  -0.0067083650    98   0.0116265649    99  -0.0037179973   101
 0.0112563375   102   0.0325136602   103   0.1892984469   104  -0.0000000001   105
 0.0647516758   107  -0.0000000000   108  -0.0070415822   110  -0.0000000002   111
 0.0029723750   113   0.0010156554   115   0.1491231290   116   0.0353714232   117
-0.0134368367   118  -0.0000000001   119
# ORBITAL         13  NAO =      90
-0.0077795017     1   0.1114492677     2   0.1676977293     3   0.0636506129     4
 0.1102407267     5   0.0070910121     7   0.0122634165     8  -0.0147424044    10
 0.0206787210    11   0.0059331824    13   0.0000881382    15   0.1838685648    16
 0.0693750950    17  -0.0071295317    18  -0.0123503755    19   0.0077795017    21
-0.1114492677    22  -0.1676977292    23   0.0636506129    24  -0.1102407270    25
 0.0070910126    27  -0.0122634173    28   0.0147424042    30   0.0206787185    31
-0.0059331821    33  -0.0000881383    35  -0.1838685644    36  -0.0693750951    37
-0.0071295320    38   0.0123503750    39  -0.0077800331    41   0.1114379546    42
 0.1677056671    43  -0.1272989969    44   0.0000000000    45  -0.0141745507    47
-0.0000000000    48   0.0162731269    50   0.0000000000    51  -0.0250837618    53
 0.0000883276    55   0.1838681097    56   0.0693765065    57   0.0142627162    58
 0.0000000000    59   0.0077795017    61  -0.1114492678    62  -0.1676977292    63
 0.0636506129    64   0.1102407272    65   0.0070910126    67   0.0122634172    68
 0.0147424042    70  -0.0206787184    71  -0.0059331821    73  -0.0000881382    75
-0.1838685642    76  -0.0693750950    77  -0.0071295322    78  -0.0123503751    79
-0.0077795017    81   0.1114492677    82   0.1676977293    83   0.0636506130    84
-0.1102407268    85   0.0070910120    87  -0.0122634164    88  -0.0147424044    90
-0.0206787211    91   0.0059331824    93   0.0000881382    95   0.1838685647    96
 0.0693750949    97  -0.0071295317    98   0.0123503756    99   0.0077800331   101
-0.1114379543   102  -0.1677056673   103  -0.1272989948   104   0.0000000001   105
-0.0141745525   107   0.0000000000   108  -0.0162731262   110   0.0000000001   111
 0.0250837603   113  -0.0000883285   115  -0.1838681106   116  -0.0693765078   117
 0.0142627144   118   0.0000000001   119
# ORBITAL         14  NAO =      90
 0.0000011888     1  -0.0000428122     2  -0.0000193406     3  -0.2396559405     4
 0.1383212588     5  -0.0833040860     7   0.0480780537     8   0.0081916407    10
 0.0054632598    11  -0.0081940522    13   0.0000021430    15  -0.0000568495    16
-0.0000165189    17  -0.0045591993    18   0.0026388067    19   0.0000011888    21
-0.0000428110    22  -0.0000193401    23   0.2396559394    24   0.1383212571    25
 0.0833040855    27   0.0480780535    28   0.0081916411    30  -0.0054632601    31
-0.0081940519    33   0.0000021439    35  -0.0000568498    36  -0.0000165188    37
 0.0045591989    38   0.0026388061    39  -0.0000000000    41  -0.0000000000    42
-0.0000000000    43  -0.0000000000    44  -0.2767352102    45  -0.0000000000    47
-0.0961986294    48  -0.0000000001    50  -0.0109202542    51   0.0000000000    53
-0.0000000001    55  -0.0000000000    56  -0.0000000000    57  -0.0000000000    58
-0.0052745169    59  -0.0000011888    61   0.0000428110    62   0.0000193401    63
-0.2396559396    64   0.1383212572    65  -0.0833040854    67   0.0480780534    68
-0.0081916410    70  -0.0054632601    71   0.0081940518    73  -0.0000021439    75
 0.0000568498    76   0.0000165188    77  -0.0045591989    78   0.0026388060    79
-0.0000011888    81   0.0000428123    82   0.0000193406    83   0.2396559404    84
 0.1383212588    85   0.0833040860    87   0.0480780537    88  -0.0081916407    90
 0.0054632598    91   0.0081940523    93  -0.0000021429    95   0.0000568495    96
 0.0000165189    97   0.0045591993    98   0.0026388067    99  -0.0000000000   101
 0.0000000001   102  -0.0000000000   103  -0.0000000000   104  -0.2767352058   105
-0.0000000000   107  -0.0961986304   108  -0.0000000000   110   0.0109202573   111
 0.0000000000   113  -0.0000000001   115  -0.0000000000   116  -0.0000000000   117
 0.0000000000   118  -0.0052745165   119
# ORBITAL         15  NAO =      90
-0.0006266032     1  -0.0382787571     2  -0.0483997291     3   0.1633248653     4
 0.1961580581     5   0.0509526263     7   0.0877847417     8  -0.0121775015    10
 0.0001460825    11   0.0084420611    13   0.0009659969    15   0.1928592179    16
 0.0682975466    17  -0.0044975288    18  -0.0126684488    19  -0.0006266032    21
-0.0382787568    22  -0.0483997291    23  -0.1633248656    24   0.1961580581    25
-0.0509526265    27   0.0877847418    28  -0.0121775014    30  -0.0001460827    31
 0.0084420612    33   0.0009659972    35   0.1928592182    36   0.0682975468    37
 0.0044975286    38  -0.0126684491    39  -0.0000000000    41  -0.0000000001    42
-0.0000000001    43  -0.0000000005    44  -0.0865394535    45  -0.0000000003    47
-0.0003977750    48  -0.0000000001    50  -0.0204636857    51   0.0000000000    53
-0.0000000000    55   0.0000000003    56   0.0000000001    57   0.0000000000    58
-0.0048836154    59   0.0006266032    61   0.0382787567    62   0.0483997289    63
 0.1633248656    64   0.1961580577    65   0.0509526263    67   0.0877847414    68
 0.0121775015    70  -0.0001460828    71  -0.0084420614    73  -0.0009659972    75
-0.1928592177    76  -0.0682975466    77  -0.0044975287    78  -0.0126684492    79
 0.0006266032    81   0.0382787572    82   0.0483997292    83  -0.1633248655    84
 0.1961580588    85  -0.0509526262    87   0.0877847417    88   0.0121775015    90
 0.0001460827    91  -0.0084420611    93  -0.0009659969    95  -0.1928592181    96
-0.0682975467    97   0.0044975289    98  -0.0126684490    99   0.0000000000   101
 0.0000000001   102   0.0000000001   103  -0.0000000005   104  -0.0865394521   105
-0.0000000001   107  -0.0003977754   108  -0.0000000001   110   0.0204636868   111
 0.0000000001   113  -0.0000000000   115  -0.0000000004   116  -0.0000000001   117
 0.0000000000   118  -0.0048836154   119
# ORBITAL         16  NAO =      90
-0.0003586348     1  -0.0220988824     2  -0.0279910890     3   0.0076641824     4
 0.1632676821     5   0.0290045303     7   0.0509421160     8   0.0106923614    10
 0.0119090275    11  -0.0128503064    13   0.0005575567    15   0.1113324509    16
 0.0394219179    17  -0.0074820090    18  -0.0044959757    19   0.0003586348    21
 0.0220988823    22   0.0279910887    23   0.0076641819    24  -0.1632676811    25
 0.0290045293    27  -0.0509421154    28  -0.0106923611    30   0.0119090285    31
 0.0128503057    33  -0.0005575565    35  -0.1113324506    36  -0.0394219177    37
-0.0074820091    38   0.0044959755    39   0.0007213416    41   0.0441838214    42
 0.0558579175    43   0.2904548744    44  -0.0000000001    45   0.1171902403    47
 0.0000000000    48  -0.0039203318    50  -0.0000000002    51   0.0082203577    53
-0.0011142396    55  -0.2227104062    56  -0.0788656304    57  -0.0152767920    58
 0.0000000001    59   0.0003586348    61   0.0220988824    62   0.0279910889    63
 0.0076641824    64   0.1632676819    65   0.0290045295    67   0.0509421156    68
-0.0106923610    70  -0.0119090285    71   0.0128503057    73  -0.0005575565    75
-0.1113324512    76  -0.0394219179    77  -0.0074820092    78  -0.0044959756    79
-0.0003586348    81  -0.0220988822    82  -0.0279910888    83   0.0076641819    84
-0.1632676816    85   0.0290045301    87  -0.0509421157    88   0.0106923615    90
-0.0119090275    91  -0.0128503065    93   0.0005575567    95   0.1113324502    96
 0.0394219176    97  -0.0074820090    98   0.0044959758    99  -0.0007213416   101
-0.0441838220   102  -0.0558579171   103   0.2904548721   104  -0.0000000001   105
 0.1171902421   107  -0.0000000000   108   0.0039203305   110  -0.0000000001   111
-0.0082203568   113   0.0011142407   115   0.2227104073   116   0.0788656318   117
-0.0152767901   118  -0.0000000001   119
# ORBITAL         17  NAO =      90
-0.0012911822     1  -0.0067578701     2   0.0010641979     3   0.2465963959     4
-0.0149150249     5   0.1010029777     7  -0.0041486050     8  -0.0213705643    10
 0.0047272710    11   0.0206865037    13   0.0003120959    15   0.1092296435    16
 0.0566451247    17   0.0032481770    18  -0.0081643699    19  -0.0012911822    21
-0.0067578703    22   0.0010641980    23  -0.2465963959    24  -0.0149150244    25
-0.1010029782    27  -0.0041486050    28  -0.0213705640    30  -0.0047272709    31
 0.0206865033    33   0.0003120959    35   0.1092296434    36   0.0566451249    37
-0.0032481772    38  -0.0081643701    39   0.0025873290    41   0.0136123011    42
-0.0020704704    43   0.2206386849    44   0.0000000000    45   0.0937752162    47
 0.0000000000    48  -0.0274275908    50  -0.0000000002    51   0.0288047866    53
-0.0006225099    55  -0.2183274679    56  -0.1132505770    57  -0.0108749007    58
 0.0000000001    59  -0.0012911822    61  -0.0067578702    62   0.0010641980    63
-0.2465963960    64   0.0149150247    65  -0.1010029780    67   0.0041486050    68
-0.0213705641    70   0.0047272710    71   0.0206865034    73   0.0003120959    75
 0.1092296434    76   0.0566451248    77  -0.0032481771    78   0.0081643701    79
-0.0012911822    81  -0.0067578700    82   0.0010641980    83   0.2465963959    84
 0.0149150247    85   0.1010029776    87   0.0041486049    88  -0.0213705644    90
-0.0047272711    91   0.0206865038    93   0.0003120960    95   0.1092296435    96
 0.0566451247    97   0.0032481769    98   0.0081643700    99   0.0025873290   101
 0.0136123017   102  -0.0020704709   103  -0.2206386828   104   0.0000000001   105
-0.0937752175   107   0.0000000001   108  -0.0274275902   110   0.0000000002   111
 0.0288047861   113  -0.0006225107   115  -0.2183274690   116  -0.1132505784   117
 0.0108748991   118   0.0000000001   119
# ORBITAL         18  NAO =      90
-0.0022422568     1  -0.0117861316     2   0.0017961586     3  -0.0149685697     4
 0.2293271422     5  -0.0041574297     7   0.0961923012     8  -0.0046743391    10
 0.0297488877    11   0.0035011315    13   0.0005353083    15   0.1890734811    16
 0.0980836424    17  -0.0081565837    18  -0.0061626988    19   0.0022422568    21
 0.0117861306    22  -0.0017961592    23  -0.0149685686    24  -0.2293271406    25
-0.0041574294    27  -0.0961923006    28   0.0046743386    30   0.0297488899    31
-0.0035011317    33  -0.0005353092    35  -0.1890734813    36  -0.0980836425    37
-0.0081565829    38   0.0061626996    39   0.0000000000    41  -0.0000000000    42
 0.0000000000    43   0.0000000001    44   0.2552406577    45   0.0000000001    47
 0.1034033602    48   0.0000000001    50   0.0215474843    51  -0.0000000000    53
 0.0000000001    55  -0.0000000000    56  -0.0000000000    57   0.0000000000    58
 0.0079649290    59  -0.0022422568    61  -0.0117861306    62   0.0017961592    63
 0.0149685685    64  -0.2293271407    65   0.0041574294    67  -0.0961923005    68
-0.0046743388    70   0.0297488900    71   0.0035011319    73   0.0005353091    75
 0.1890734813    76   0.0980836425    77   0.0081565830    78   0.0061626997    79
 0.0022422568    81   0.0117861315    82  -0.0017961586    83   0.0149685698    84
 0.2293271424    85   0.0041574298    87   0.0961923012    88   0.0046743391    90
 0.0297488876    91  -0.0035011315    93  -0.0005353084    95  -0.1890734810    96
-0.0980836423    97   0.0081565838    98  -0.0061626989    99   0.0000000000   101
 0.0000000000   102  -0.0000000000   103  -0.0000000000   104  -0.2552406531   105
-0.0000000000   107  -0.1034033615   108  -0.0000000001   110   0.0215474869   111
 0.0000000001   113  -0.0000000001   115  -0.0000000001   116  -0.0000000000   117
-0.0000000000   118  -0.0079649287   119
# ORBITAL         19  NAO =       5
-0.6069137242     6  -0.5288219485     9   0.0160778803    12   0.0278520511    14
-0.0182054019    20
# ORBITAL         20  NAO =       5
-0.6069137241    26  -0.5288219486    29  -0.0160778803    32   0.0278520509    34
-0.0182054020    40
# ORBITAL         21  NAO =       5
-0.6069128657    46  -0.5288236221    49  -0.0321586274    52   0.0000000002    54
-0.0182031846    60
# ORBITAL         22  NAO =       5
-0.6069137243    66  -0.5288219484    69  -0.0160778802    72  -0.0278520508    74
-0.0182054020    80
# ORBITAL         23  NAO =       5
-0.6069137243    86  -0.5288219484    89   0.0160778802    92  -0.0278520511    94
-0.0182054019   100
# ORBITAL         24  NAO =       5
-0.6069128633   106  -0.5288236244   109   0.0321586294   112   0.0000000002   114
-0.0182031848   120
$end

In the input file, VB structures are specified by the users rather than automatically generated. The keyword NSTR=2 in $CTRL section specifies 2 VB structures in the computation, and the 2 Kekulé structures are described in section $STR. The keyword GUESS=READ means that the initial guess is obtained by the user in $GUS section. The initial guess comes from the optmized orbitals of previous computation, stored in corresponding ORB file.

2.2.3.2. Computational Results

The final energy of the computation can be found after VBSCF iterations as -230.62995 hartree. Compared with the energy including 5 covalent structures, the corresponding resonance energy from Dewar structures can be obtained as 2.8 kcal/mol.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 C           2 H            1.093       0.979
 1 C           3 C            1.399       1.186
 3 C           4 H            1.093       0.979
 3 C           5 C            1.399       1.186
 5 C           6 H            1.093       0.979
 5 C           7 C            1.399       1.186
 7 C           8 H            1.093       0.979
 7 C           9 C            1.399       1.186
 9 C          10 H            1.093       0.979
 1 C          11 C            1.399       1.186
 9 C          11 C            1.399       1.186
11 C          12 H            1.093       0.979

The atomic population analysis also shows that the bonder orders of CC bondings are the same as 1.186. Compared with the bond order from 5 covalent structures and the resonance energy from Dewar structures, the conjugation and stability mainly come from the resonance of Kekulé structures.

2.2.4. Computations with 1 Kekulé Structure

Finally, the computation including only 1 Kekulé structure, which represents a cycloalkene, will be proceeded to check the resonance from Kekulé structures.

2.2.4.1. Input File

The input file of this computation is shown below. The initial guess comes from the optimized orbitals of the computation including 2 Kekulé structures.

C6H6
$ctrl
nstr=1 nao=6 nae=6 iscf=5 iprint=3
orbtyp=hao frgtyp=sao
int=libcint basis=cc-pvdz
guess=read
$end
$str
1:18 19-24
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
 C           0.6995000584   1.2115696411   0.0000000000
 H           1.2460106991   2.1581538376   0.0000000000
 C          -0.6995000584   1.2115696411   0.0000000000
 H          -1.2460106991   2.1581538376   0.0000000000
 C          -1.3990001169   0.0000000000   0.0000000000
 H          -2.4920213982   0.0000000000   0.0000000000
 C          -0.6995000584  -1.2115696411   0.0000000000
 H          -1.2460106991  -2.1581538376   0.0000000000
 C           0.6995000584  -1.2115696411   0.0000000000
 H           1.2460106991  -2.1581538376   0.0000000000
 C           1.3990001169   0.0000000000   0.0000000000
 H           2.4920213982   0.0000000000   0.0000000000
$end
$gus
    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90    90     5     5     5     5     5     5
# ORBITAL          1  NAO =      90
-0.4085796783     1  -0.0018719283     2   0.0001665818     3   0.0001010685     4
 0.0001981258     5   0.0000450438     7   0.0000863179     8   0.0009553396    10
 0.0000836069    11   0.0010374634    13   0.0011387900    15  -0.0001957720    16
-0.0001340753    17   0.0000469703    18   0.0000827563    19  -0.4085796727    21
-0.0018719292    22   0.0001665812    23  -0.0001010727    24   0.0001981248    25
-0.0000450425    27   0.0000863186    28   0.0009553426    30  -0.0000836019    31
 0.0010374618    33   0.0011387887    35  -0.0001957726    36  -0.0001340767    37
-0.0000469715    38   0.0000827567    39  -0.4090835453    41  -0.0018962542    42
 0.0001483676    43  -0.0002144425    44   0.0000004419    45  -0.0001064444    47
 0.0000000620    48   0.0010795224    50  -0.0000005324    51   0.0009089949    53
 0.0011398054    55  -0.0002140828    56  -0.0001424123    57  -0.0000967357    58
 0.0000004933    59  -0.4085796762    61  -0.0018717962    62   0.0001665571    63
-0.0001012559    64  -0.0001976491    65  -0.0000451304    67  -0.0000860262    68
 0.0009554043    70   0.0000853899    71   0.0010374754    73   0.0011388868    75
-0.0001954373    76  -0.0001341537    77  -0.0000475649    78  -0.0000822130    79
-0.4085796747    81  -0.0018717874    82   0.0001665574    83   0.0001012488    84
-0.0001976456    85   0.0000451252    87  -0.0000860254    88   0.0009553937    90
-0.0000853972    91   0.0010374881    93   0.0011388931    95  -0.0001954365    96
-0.0001341565    97   0.0000475538    98  -0.0000822115    99  -0.4090835098   101
-0.0018962560   102   0.0001483691   103   0.0002144405   104   0.0000004455   105
 0.0001064418   107   0.0000000681   108   0.0010795349   110   0.0000005509   111
 0.0009089845   113   0.0011398035   115  -0.0002140768   116  -0.0001424127   117
 0.0000967353   118   0.0000005091   119
# ORBITAL          2  NAO =      90
 0.2888921826     1   0.0012283761     2  -0.0002515233     3   0.0001334436     4
-0.0001583616     5   0.0001866742     7  -0.0001486589     8  -0.0008194761    10
-0.0001545655    11  -0.0005903612    13  -0.0007993769    15   0.0001561841    16
 0.0000815611    17   0.0000143387    18  -0.0000796338    19  -0.2888922723    21
-0.0012282896    22   0.0002516068    23   0.0001334933    24   0.0001581461    25
 0.0001866825    27   0.0001482790    28   0.0008195384    30  -0.0001557097    31
 0.0005903305    33   0.0007993976    35  -0.0001560171    36  -0.0000815893    37
 0.0000140002    38   0.0000793106    39  -0.5781886874    41  -0.0024517953    42
 0.0004960627    43  -0.0001410124    44   0.0000000144    45  -0.0000703152    47
 0.0000000565    48   0.0015346481    50  -0.0000001631    51   0.0013051183    53
 0.0016011118    55  -0.0003076892    56  -0.0001586915    57  -0.0001240234    58
 0.0000001386    59  -0.2888922230    61  -0.0012282282    62   0.0002515995    63
 0.0001335274    64  -0.0001581270    65   0.0001867496    67  -0.0001482931    68
 0.0008194516    70   0.0001556375    71   0.0005904439    73   0.0007994395    75
-0.0001560266    76  -0.0000816204    77   0.0000141380    78  -0.0000793592    79
 0.2888922537    81   0.0012283054    82  -0.0002515181    83   0.0001334812    84
 0.0001583237    85   0.0001867191    87   0.0001486497    88  -0.0008194148    90
 0.0001546034    91  -0.0005904524    93  -0.0007994211    95   0.0001561754    96
 0.0000815940    97   0.0000144233    98   0.0000796423    99   0.5781886588   101
 0.0024515577   102  -0.0004959802   103  -0.0001409979   104  -0.0000000307   105
-0.0000705453   107  -0.0000000454   108  -0.0015343653   110  -0.0000001509   111
-0.0013054893   113  -0.0016012734   115   0.0003078240   116   0.0001587913   117
-0.0001239620   118  -0.0000001283   119
# ORBITAL          3  NAO =      90
 0.5006101737     1   0.0021131041     2  -0.0004386602     3  -0.0001720820     4
-0.0000537351     5  -0.0001544833     7   0.0000103521     8  -0.0010913048    10
-0.0000446596    11  -0.0013590329    13  -0.0013847882    15   0.0002530053    16
 0.0001361579    17  -0.0000788740    18  -0.0000770243    19   0.5006101452    21
 0.0021129625    22  -0.0004386520    23   0.0001720253    24  -0.0000536574    25
 0.0001543765    27   0.0000103495    28  -0.0010911679    30   0.0000446577    31
-0.0013592283    33  -0.0013848833    35   0.0002530062    36   0.0001362215    37
 0.0000786944    38  -0.0000770459    39  -0.0000000492    41  -0.0000000438    42
 0.0000000133    43  -0.0000000140    44   0.0002083475    45   0.0000000368    47
 0.0002656048    48   0.0000000488    50   0.0002187797    51  -0.0000000628    53
-0.0000000239    55   0.0000000279    56   0.0000000168    57  -0.0000000076    58
 0.0000604882    59  -0.5006102068    61  -0.0021129621    62   0.0004386735    63
-0.0001720307    64  -0.0000536122    65  -0.0001543752    67   0.0000104714    68
 0.0010911805    70   0.0000450132    71   0.0013592249    73   0.0013848909    75
-0.0002529465    76  -0.0001362288    77  -0.0000787963    78  -0.0000769463    79
-0.5006101688    81  -0.0021131026    82   0.0004386817    83   0.0001720878    84
-0.0000536878    85   0.0001544821    87   0.0000104746    88   0.0010913171    90
-0.0000450214    91   0.0013590299    93   0.0013847960    95  -0.0002529452    96
-0.0001361656    97   0.0000789791    98  -0.0000769196    99   0.0000000225   101
-0.0000000446   102   0.0000000134   103   0.0000000135   104   0.0002082166   105
-0.0000000368   107   0.0002654040   108   0.0000000506   110  -0.0002194550   111
-0.0000000653   113  -0.0000000248   115   0.0000000293   116   0.0000000173   117
 0.0000000083   118   0.0000599174   119
# ORBITAL          4  NAO =      90
-0.2893480425     1  -0.0011811095     2   0.0006744406     3   0.0001189419     4
-0.0000446067     5   0.0000569570     7  -0.0002197480     8   0.0005592017    10
-0.0000156074    11   0.0006968256    13   0.0007625480    15  -0.0000777904    16
-0.0000296491    17   0.0000238435    18   0.0000028641    19  -0.2893481096    21
-0.0011811399    22   0.0006744442    23  -0.0001189574    24  -0.0000445949    25
-0.0000569839    27  -0.0002197505    28   0.0005592352    30   0.0000155929    31
 0.0006967797    33   0.0007625291    35  -0.0000777899    36  -0.0000296344    37
-0.0000238983    38   0.0000028476    39   0.5779780421    41   0.0024236036    42
-0.0013095349    43  -0.0000033117    44  -0.0000000730    45  -0.0003415559    47
 0.0000000126    48  -0.0012987191    50   0.0000000214    51  -0.0012017051    53
-0.0015230608    55   0.0002233907    56   0.0000742468    57   0.0000352458    58
-0.0000000253    59  -0.2893480526    61  -0.0011811337    62   0.0006744475    63
-0.0001189109    64   0.0000445154    65  -0.0000569376    67   0.0002196894    68
 0.0005591878    70  -0.0000159673    71   0.0006968259    73   0.0007625328    75
-0.0000778596    76  -0.0000296337    77  -0.0000237314    78  -0.0000029671    79
-0.2893481027    81  -0.0011811672    82   0.0006744484    83   0.0001189251    84
 0.0000444960    85   0.0000569605    87   0.0002196888    88   0.0005592200    90
 0.0000159731    91   0.0006967804    93   0.0007625097    95  -0.0000778598    96
-0.0000296196    97   0.0000237689    98  -0.0000029652    99   0.5779780591   101
 0.0024236073   102  -0.0013095365   103   0.0000033130   104  -0.0000000990   105
 0.0003415570   107  -0.0000000336   108  -0.0012987290   110  -0.0000001709   111
-0.0012016972   113  -0.0015230594   115   0.0002233876   116   0.0000742455   117
-0.0000352451   118  -0.0000001515   119
# ORBITAL          5  NAO =      90
-0.5007499006     1  -0.0020932788     2   0.0011339536     3  -0.0000684881     4
 0.0000697482     5  -0.0002271535     7  -0.0001934040     8   0.0010981889    10
 0.0000685550    11   0.0010726975    13   0.0013215420    15  -0.0001608701    16
-0.0000525753    17   0.0000029695    18   0.0000308210    19   0.5007498476    21
 0.0020932696    22  -0.0011339307    23  -0.0000684705    24  -0.0000697471    25
-0.0002271239    27   0.0001933327    28  -0.0010982033    30   0.0000684523    31
-0.0010726842    33  -0.0013215463    35   0.0001608813    36   0.0000525762    37
 0.0000029679    38  -0.0000308358    39   0.0000000362    41  -0.0000000393    42
 0.0000000128    43  -0.0000000095    44   0.0001873871    45   0.0000000353    47
 0.0002028610    48   0.0000000422    50   0.0000930299    51  -0.0000000564    53
-0.0000000238    55   0.0000000220    56   0.0000000158    57  -0.0000000073    58
 0.0000255403    59  -0.5007498869    61  -0.0020932596    62   0.0011339537    63
 0.0000684717    64  -0.0000696927    65   0.0002271254    67   0.0001934495    68
 0.0010982216    70   0.0000687936    71   0.0010726748    73   0.0013215517    75
-0.0001608273    76  -0.0000525822    77  -0.0000030701    78  -0.0000307435    79
 0.5007498440    81   0.0020932955    82  -0.0011339306    83   0.0000684810    84
 0.0000698075    85   0.0002271480    87  -0.0001932870    88  -0.0010981803    90
 0.0000681955    91  -0.0010726973    93  -0.0013215327    95   0.0001609262    96
 0.0000525662    97  -0.0000028726    98   0.0000309235    99   0.0000000242   101
-0.0000000429   102   0.0000000121   103   0.0000000074   104  -0.0001872809   105
-0.0000000363   107  -0.0002028699   108   0.0000000512   110   0.0000931069   111
-0.0000000660   113  -0.0000000272   115   0.0000000283   116   0.0000000166   117
 0.0000000105   118  -0.0000254667   119
# ORBITAL          6  NAO =      90
 0.4091242805     1   0.0018190810     2  -0.0036449243     3   0.0000003109     4
-0.0000029945     5   0.0007740556     7   0.0013380065     8  -0.0007544593    10
 0.0000780477    11  -0.0006748974    13  -0.0009960142    15  -0.0000423640    16
-0.0002163202    17   0.0000245090    18   0.0000435846    19  -0.4091242314    21
-0.0018191745    22   0.0036448420    23   0.0000002969    24   0.0000032856    25
 0.0007740526    27  -0.0013375548    28   0.0007543969    30   0.0000795334    31
 0.0006749172    33   0.0009959764    35   0.0000421336    36   0.0002163607    37
 0.0000249463    38  -0.0000431647    39   0.4088374571    41   0.0018140837    42
-0.0036448324    43   0.0000011734    44   0.0000001409    45  -0.0015470931    47
 0.0000001187    48  -0.0006363791    50  -0.0000004489    51  -0.0007950336    53
-0.0009967424    55  -0.0000464723    56  -0.0002169062    57  -0.0000506419    58
 0.0000003853    59  -0.4091242365    61  -0.0018190027    62   0.0036448289    63
 0.0000003326    64  -0.0000031321    65   0.0007741531    67   0.0013375953    68
 0.0007542579    70  -0.0000792764    71   0.0006751316    73   0.0009960980    75
 0.0000421797    76   0.0002162805    77   0.0000250537    78   0.0000432185    79
 0.4091242781    81   0.0018189398    82  -0.0036449221    83   0.0000004009    84
 0.0000029779    85   0.0007741877    87  -0.0013379594    88  -0.0007542979    90
-0.0000783208    91  -0.0006751119    93  -0.0009961062    95  -0.0000423138    96
-0.0002162634    97   0.0000247982    98  -0.0000434744    99  -0.4088374402   101
-0.0018138102   102   0.0036447434   103   0.0000011435   104  -0.0000000067   105
-0.0015468397   107  -0.0000001002   108   0.0006360602   110  -0.0000002972   111
 0.0007954525   113   0.0009969239   115   0.0000463126   116   0.0002167944   117
-0.0000507097   118  -0.0000002434   119
# ORBITAL          7  NAO =      90
-0.0105310908     1   0.1791319489     2   0.0923971303     3  -0.0318962677     4
-0.0553214746     5  -0.0062990419     7  -0.0109406130     8   0.0108261423    10
-0.0038130194    11   0.0070179874    13  -0.0063573590    15   0.0456220599    16
 0.0036425281    17  -0.0036155711    18  -0.0062523687    19  -0.0105314089    21
 0.1791319474    22   0.0923971349    23   0.0318962676    24  -0.0553214745    25
 0.0062990427    27  -0.0109406141    28   0.0108261425    30   0.0038130195    31
 0.0070179875    33  -0.0063573581    35   0.0456220596    36   0.0036425283    37
 0.0036155718    38  -0.0062523691    39  -0.0106286110    41   0.1791775670    42
 0.0924119143    43   0.0638644527    44  -0.0000000012    45   0.0126250153    47
-0.0000000001    48   0.0051125578    50   0.0000000011    51   0.0127368885    53
-0.0063570874    55   0.0456280497    56   0.0036351310    57   0.0072220519    58
-0.0000000009    59  -0.0105311113    61   0.1791319485    62   0.0923971342    63
 0.0318962682    64   0.0553214734    65   0.0062990428    67   0.0109406133    68
 0.0108261418    70  -0.0038130232    71   0.0070179867    73  -0.0063573593    75
 0.0456220589    76   0.0036425285    77   0.0036155729    78   0.0062523682    79
-0.0105311036    81   0.1791319484    82   0.0923971305    83  -0.0318962682    84
 0.0553214734    85  -0.0062990421    87   0.0109406123    88   0.0108261422    90
 0.0038130230    91   0.0070179872    93  -0.0063573596    95   0.0456220592    96
 0.0036425281    97  -0.0036155721    98   0.0062523679    99  -0.0106297497   101
 0.1791775616   102   0.0924119206   103  -0.0638644524   104  -0.0000000012   105
-0.0126250177   107  -0.0000000000   108   0.0051125593   110  -0.0000000010   111
 0.0127368898   113  -0.0063570850   115   0.0456280500   116   0.0036351304   117
-0.0072220520   118  -0.0000000009   119
# ORBITAL          8  NAO =      90
 0.0076028895     1  -0.1247833094     2  -0.0702498150     3  -0.0785132216     4
 0.0639777661     5  -0.0281449519     7   0.0270091285     8   0.0056384214    10
 0.0119231520    11  -0.0143970361    13   0.0039295681    15  -0.0517335466    16
-0.0069559347    17   0.0014213923    18   0.0074921988    19  -0.0076028888    21
 0.1247833182    22   0.0702498194    23  -0.0785132193    24  -0.0639777657    25
-0.0281449519    27  -0.0270091281    28  -0.0056384209    30   0.0119231587    31
 0.0143970362    33  -0.0039295685    35   0.0517335493    36   0.0069559353    37
 0.0014213952    38  -0.0074922001    39  -0.0151821569    41   0.2495710190    42
 0.1404874823    43   0.0322982227    44   0.0000000019    45   0.0186231256    47
 0.0000000007    48   0.0009086803    50   0.0000000010    51   0.0166300085    53
-0.0078591029    55   0.1034526736    56   0.0139125616    57   0.0143887842    58
-0.0000000004    59  -0.0076028851    61   0.1247833096    62   0.0702498147    63
-0.0785132218    64   0.0639777655    65  -0.0281449531    67   0.0270091281    68
-0.0056384214    70  -0.0119231586    71   0.0143970359    73  -0.0039295685    75
 0.0517335459    76   0.0069559349    77   0.0014213945    78   0.0074921998    79
 0.0076028868    81  -0.1247833174    82  -0.0702498196    83  -0.0785132191    84
-0.0639777655    85  -0.0281449509    87  -0.0270091286    88   0.0056384206    90
-0.0119231525    91  -0.0143970358    93   0.0039295688    95  -0.0517335500    96
-0.0069559352    97   0.0014213925    98  -0.0074921991    99   0.0151821472   101
-0.2495710187   102  -0.1404874829   103   0.0322982228   104   0.0000000027   105
 0.0186231264   107   0.0000000008   108  -0.0009086819   110   0.0000000004   111
-0.0166300072   113   0.0078591049   115  -0.1034526738   116  -0.0139125612   117
 0.0143887828   118   0.0000000005   119
# ORBITAL          9  NAO =      90
 0.0131251465     1  -0.2161385559     2  -0.1216727085     3   0.0639730896     4
-0.0046068035     5   0.0270059763     7   0.0030477106     8  -0.0179156928    10
 0.0021926302    11   0.0027320071    13   0.0068069785    15  -0.0895734340    16
-0.0120431889    17   0.0074864530    18   0.0100635578    19   0.0131251217    21
-0.2161385511    22  -0.1216727056    23  -0.0639730923    24  -0.0046068060    25
-0.0270059769    27   0.0030477096    28  -0.0179156936    30  -0.0021926297    31
 0.0027320083    33   0.0068069785    35  -0.0895734318    36  -0.0120431888    37
-0.0074864525    38   0.0100635578    39  -0.0000000465    41   0.0000000049    42
 0.0000000028    43   0.0000000007    44  -0.1154806636    45   0.0000000001    47
-0.0437490908    48  -0.0000000002    50  -0.0184557093    51   0.0000000009    53
 0.0000000000    55   0.0000000019    56   0.0000000002    57   0.0000000004    58
-0.0029085310    59  -0.0131252134    61   0.2161385558    62   0.1216727087    63
 0.0639730895    64  -0.0046068039    65   0.0270059757    67   0.0030477102    68
 0.0179156937    70  -0.0021926324    71  -0.0027320077    73  -0.0068069787    75
 0.0895734335    76   0.0120431889    77   0.0074864531    78   0.0100635579    79
-0.0131251919    81   0.2161385514    82   0.1216727057    83  -0.0639730926    84
-0.0046068062    85  -0.0270059772    87   0.0030477091    88   0.0179156935    90
 0.0021926320    91  -0.0027320077    93  -0.0068069785    95   0.0895734316    96
 0.0120431885    97  -0.0074864534    98   0.0100635574    99  -0.0000000629   101
-0.0000000047   102  -0.0000000024   103   0.0000000005   104  -0.1154806635   105
 0.0000000006   107  -0.0437490900   108  -0.0000000003   110   0.0184557112   111
 0.0000000003   113   0.0000000003   115  -0.0000000021   116  -0.0000000004   117
 0.0000000002   118  -0.0029085291   119
# ORBITAL         10  NAO =      90
-0.0049057145     1   0.1006640065     2   0.0846904205     3  -0.1397564545     4
 0.1141784842     5  -0.0456691919     7   0.0234357582     8   0.0117723759    10
 0.0056723402    11  -0.0100538947    13  -0.0019191218    15   0.0770189499    16
 0.0164355039    17  -0.0072749056    18  -0.0060278708    19  -0.0049056971    21
 0.1006640036    22   0.0846904176    23   0.1397564569    24   0.1141784844    25
 0.0456691926    27   0.0234357589    28   0.0117723756    30  -0.0056723405    31
-0.0100538947    33  -0.0019191215    35   0.0770189473    36   0.0164355032    37
 0.0072749056    38  -0.0060278707    39   0.0099053219    41  -0.2012723368    42
-0.1693914578    43   0.0583006645    44  -0.0000000021    45  -0.0049813010    47
-0.0000000006    48   0.0007093782    50  -0.0000000002    51  -0.0041287655    53
 0.0038346686    55  -0.1542323428    56  -0.0329233571    57  -0.0177384123    58
-0.0000000001    59  -0.0049057294    61   0.1006640075    62   0.0846904208    63
 0.1397564543    64  -0.1141784844    65   0.0456691920    67  -0.0234357583    68
 0.0117723757    70   0.0056723403    71  -0.0100538948    73  -0.0019191218    75
 0.0770189499    76   0.0164355037    77   0.0072749057    78   0.0060278712    79
-0.0049057266    81   0.1006640034    82   0.0846904175    83  -0.1397564568    84
-0.1141784844    85  -0.0456691923    87  -0.0234357585    88   0.0117723760    90
-0.0056723404    91  -0.0100538949    93  -0.0019191216    95   0.0770189470    96
 0.0164355031    97  -0.0072749055    98   0.0060278709    99   0.0099053293   101
-0.2012723391   102  -0.1693914579   103  -0.0583006644   104   0.0000000023   105
 0.0049813014   107   0.0000000005   108   0.0007093780   110   0.0000000002   111
-0.0041287656   113   0.0038346685   115  -0.1542323426   116  -0.0329233566   117
 0.0177384131   118   0.0000000002   119
# ORBITAL         11  NAO =      90
-0.0085816245     1   0.1743307075     2   0.1466700542     3   0.1142616239     4
-0.0078017212     5   0.0234796112     7  -0.0185688373     8  -0.0034134122    10
-0.0060578410    11   0.0063987626    13  -0.0033214565    15   0.1334722320    16
 0.0284803972    17  -0.0060295874    18  -0.0142387079    19   0.0085817066    21
-0.1743307092    22  -0.1466700558    23   0.1142616215    24   0.0078017193    25
 0.0234796105    27   0.0185688370    28   0.0034134117    30  -0.0060578404    31
-0.0063987626    33   0.0033214561    35  -0.1334722334    36  -0.0284803975    37
-0.0060295876    38   0.0142387083    39   0.0000000232    41   0.0000000025    42
 0.0000000015    43  -0.0000000006    44  -0.2057189432    45  -0.0000000001    47
-0.0592113756    48  -0.0000000002    50  -0.0158769336    51   0.0000000002    53
-0.0000000002    55   0.0000000016    56   0.0000000002    57   0.0000000002    58
-0.0037955897    59  -0.0085818294    61   0.1743307067    62   0.1466700548    63
-0.1142616247    64   0.0078017211    65  -0.0234796110    67   0.0185688378    68
-0.0034134117    70  -0.0060578412    71   0.0063987630    73  -0.0033214559    75
 0.1334722315    76   0.0284803970    77   0.0060295877    78   0.0142387083    79
 0.0085817260    81  -0.1743307096    82  -0.1466700567    83  -0.1142616213    84
-0.0078017187    85  -0.0234796100    87  -0.0185688372    88   0.0034134116    90
-0.0060578408    91  -0.0063987625    93   0.0033214565    95  -0.1334722335    96
-0.0284803973    97   0.0060295877    98  -0.0142387085    99   0.0000000049   101
 0.0000000018   102   0.0000000020   103   0.0000000006   104   0.2057189443   105
-0.0000000002   107   0.0592113755   108   0.0000000002   110  -0.0158769330   111
-0.0000000001   113   0.0000000002   115   0.0000000015   116   0.0000000004   117
-0.0000000002   118   0.0037955897   119
# ORBITAL         12  NAO =      90
-0.0036804733     1   0.0117300772     2   0.0327949959     3   0.0945818191     4
 0.1639064451     5   0.0323475941     7   0.0560232362     8   0.0005071356    10
-0.0050238617    11  -0.0045215349    13   0.0010077685    15   0.1492702225    16
 0.0353806840    17  -0.0067287801    18  -0.0116586245    19  -0.0036801487    21
 0.0117300780    22   0.0327949909    23  -0.0945818188    24   0.1639064458    25
-0.0323475949    27   0.0560232372    28   0.0005071356    30   0.0050238619    31
-0.0045215352    33   0.0010077677    35   0.1492702221    36   0.0353806834    37
 0.0067287796    38  -0.0116586240    39  -0.0037706044    41   0.0115958520    42
 0.0326192354    43  -0.1891354471    44  -0.0000000001    45  -0.0646757241    47
 0.0000000000    48  -0.0070415642    50   0.0000000004    51   0.0030215816    53
 0.0010143114    55   0.1491037933    56   0.0353238319    57   0.0134440798    58
-0.0000000003    59  -0.0036804485    61   0.0117300767    62   0.0327949914    63
-0.0945818192    64  -0.1639064466    65  -0.0323475947    67  -0.0560232367    68
 0.0005071362    70  -0.0050238629    71  -0.0045215347    73   0.0010077684    75
 0.1492702215    76   0.0353806829    77   0.0067287799    78   0.0116586241    79
-0.0036804586    81   0.0117300775    82   0.0327949955    83   0.0945818193    84
-0.1639064457    85   0.0323475941    87  -0.0560232360    88   0.0005071357    90
 0.0050238628    91  -0.0045215349    93   0.0010077686    95   0.1492702219    96
 0.0353806835    97  -0.0067287807    98   0.0116586245    99  -0.0037694668   101
 0.0115958596   102   0.0326192295   103   0.1891354472   104  -0.0000000001   105
 0.0646757261   107   0.0000000000   108  -0.0070415657   110  -0.0000000006   111
 0.0030215803   113   0.0010143087   115   0.1491037928   116   0.0353238322   117
-0.0134440812   118  -0.0000000004   119
# ORBITAL         13  NAO =      90
-0.0075512224     1   0.1114726348     2   0.1678599129     3   0.0636737890     4
 0.1102660175     5   0.0071157447     7   0.0123065687     8  -0.0147668118    10
 0.0206625187    11   0.0059162564    13   0.0000670171    15   0.1838227411    16
 0.0692803796    17  -0.0071326782    18  -0.0123576817    19   0.0075512222    21
-0.1114726347    22  -0.1678599129    23   0.0636737893    24  -0.1102660177    25
 0.0071157446    27  -0.0123065681    28   0.0147668120    30   0.0206625205    31
-0.0059162565    33  -0.0000670170    35  -0.1838227417    36  -0.0692803795    37
-0.0071326779    38   0.0123576821    39  -0.0075377798    41   0.1114776373    42
 0.1678699600    43  -0.1273340648    44   0.0000000002    45  -0.0142192259    47
 0.0000000000    48   0.0162548947    50  -0.0000000003    51  -0.0250878035    53
 0.0000679682    55   0.1838193396    56   0.0692809136    57   0.0142604356    58
 0.0000000003    59   0.0075512263    61  -0.1114726350    62  -0.1678599130    63
 0.0636737898    64   0.1102660184    65   0.0071157445    67   0.0123065680    68
 0.0147668119    70  -0.0206625199    71  -0.0059162563    73  -0.0000670168    75
-0.1838227412    76  -0.0692803792    77  -0.0071326782    78  -0.0123576825    79
-0.0075512249    81   0.1114726350    82   0.1678599130    83   0.0636737895    84
-0.1102660178    85   0.0071157447    87  -0.0123065685    88  -0.0147668117    90
-0.0206625190    91   0.0059162561    93   0.0000670170    95   0.1838227409    96
 0.0692803794    97  -0.0071326781    98   0.0123576822    99   0.0075377838   101
-0.1114776371   102  -0.1678699601   103  -0.1273340659   104   0.0000000001   105
-0.0142192253   107  -0.0000000000   108  -0.0162548953   110   0.0000000000   111
 0.0250878041   113  -0.0000679680   115  -0.1838193399   116  -0.0692809135   117
 0.0142604366   118  -0.0000000000   119
# ORBITAL         14  NAO =      90
-0.0000242668     1   0.0000211338     2   0.0000197397     3  -0.2396956057     4
 0.1383354950     5  -0.0832832747     7   0.0480626417     8   0.0082077199    10
 0.0054692760    11  -0.0082072207    13  -0.0000008338    15  -0.0000306106    16
-0.0000082652    17  -0.0045627706    18   0.0026348910    19  -0.0000242928    21
 0.0000211338    22   0.0000197399    23   0.2396956060    24   0.1383354958    25
 0.0832832749    27   0.0480626416    28   0.0082077197    30  -0.0054692758    31
-0.0082072205    33  -0.0000008342    35  -0.0000306103    36  -0.0000082653    37
 0.0045627709    38   0.0026348912    39   0.0000000316    41   0.0000000004    42
 0.0000000001    43   0.0000000000    44  -0.2767109921    45   0.0000000000    47
-0.0961484641    48  -0.0000000002    50  -0.0109288348    51  -0.0000000001    53
-0.0000000002    55   0.0000000000    56   0.0000000000    57   0.0000000000    58
-0.0052653392    59   0.0000243486    61  -0.0000211336    62  -0.0000197401    63
-0.2396956060    64   0.1383354959    65  -0.0832832749    67   0.0480626415    68
-0.0082077198    70  -0.0054692758    71   0.0082072205    73   0.0000008342    75
 0.0000306102    76   0.0000082652    77  -0.0045627709    78   0.0026348913    79
 0.0000243167    81  -0.0000211335    82  -0.0000197398    83   0.2396956060    84
 0.1383354953    85   0.0832832746    87   0.0480626416    88  -0.0082077199    90
 0.0054692764    91   0.0082072205    93   0.0000008337    95   0.0000306105    96
 0.0000082651    97   0.0045627706    98   0.0026348910    99   0.0000000566   101
 0.0000000007   102  -0.0000000000   103  -0.0000000000   104  -0.2767109937   105
 0.0000000000   107  -0.0961484638   108  -0.0000000005   110   0.0109288338   111
 0.0000000001   113  -0.0000000005   115   0.0000000001   116  -0.0000000001   117
-0.0000000000   118  -0.0052653392   119
# ORBITAL         15  NAO =      90
-0.0009983767     1  -0.0386214684     2  -0.0485455904     3   0.1633421841     4
 0.1962473237     5   0.0509807373     7   0.0877733678     8  -0.0121832761    10
 0.0001162009    11   0.0084147730    13   0.0010223542    15   0.1927411229    16
 0.0682210653    17  -0.0045132574    18  -0.0126863282    19  -0.0009983769    21
-0.0386214683    22  -0.0485455906    23  -0.1633421842    24   0.1962473245    25
-0.0509807372    27   0.0877733679    28  -0.0121832762    30  -0.0001162008    31
 0.0084147732    33   0.0010223542    35   0.1927411233    36   0.0682210653    37
 0.0045132578    38  -0.0126863281    39   0.0000000963    41   0.0000000005    42
 0.0000000001    43  -0.0000000002    44  -0.0868168628    45  -0.0000000004    47
-0.0005743185    48  -0.0000000005    50  -0.0204836282    51   0.0000000001    53
-0.0000000004    55   0.0000000002    56   0.0000000002    57   0.0000000001    58
-0.0048842722    59   0.0009985499    61   0.0386214692    62   0.0485455904    63
 0.1633421850    64   0.1962473253    65   0.0509807366    67   0.0877733671    68
 0.0121832762    70  -0.0001162013    71  -0.0084147740    73  -0.0010223547    75
-0.1927411227    76  -0.0682210647    77  -0.0045132580    78  -0.0126863286    79
 0.0009985050    81   0.0386214693    82   0.0485455908    83  -0.1633421847    84
 0.1962473254    85  -0.0509807370    87   0.0877733675    88   0.0121832760    90
 0.0001162015    91  -0.0084147736    93  -0.0010223547    95  -0.1927411232    96
-0.0682210651    97   0.0045132576    98  -0.0126863286    99   0.0000001212   101
 0.0000000009   102  -0.0000000000   103  -0.0000000009   104  -0.0868168632   105
 0.0000000001   107  -0.0005743183   108  -0.0000000006   110   0.0204836287   111
 0.0000000000   113  -0.0000000004   115  -0.0000000005   116  -0.0000000001   117
-0.0000000000   118  -0.0048842716   119
# ORBITAL         16  NAO =      90
-0.0005894251     1  -0.0223088039     2  -0.0280654665     3   0.0075643177     4
 0.1633898810     5   0.0288954659     7   0.0510111520     8   0.0106960596    10
 0.0118868073    11  -0.0128757602    13   0.0005926770    15   0.1112764638    16
 0.0393823208    17  -0.0074782289    18  -0.0045126110    19   0.0005894222    21
 0.0223088033    22   0.0280654662    23   0.0075643166    24  -0.1633898807    25
 0.0288954662    27  -0.0510111513    28  -0.0106960596    30   0.0118868066    31
 0.0128757604    33  -0.0005926768    35  -0.1112764634    36  -0.0393823205    37
-0.0074782291    38   0.0045126111    39   0.0011830149    41   0.0446313651    42
 0.0560606696    43   0.2905583224    44   0.0000000000    45   0.1172071367    47
 0.0000000000    48  -0.0038785631    50  -0.0000000003    51   0.0082267364    53
-0.0011834653    55  -0.2225345572    56  -0.0787777804    57  -0.0152898732    58
 0.0000000001    59   0.0005894244    61   0.0223088033    62   0.0280654664    63
 0.0075643175    64   0.1633898819    65   0.0288954663    67   0.0510111514    68
-0.0106960595    70  -0.0118868065    71   0.0128757604    73  -0.0005926768    75
-0.1112764638    76  -0.0393823205    77  -0.0074782294    78  -0.0045126114    79
-0.0005894255    81  -0.0223088036    82  -0.0280654663    83   0.0075643174    84
-0.1633898808    85   0.0288954658    87  -0.0510111516    88   0.0106960598    90
-0.0118868073    91  -0.0128757604    93   0.0005926770    95   0.1112764630    96
 0.0393823204    97  -0.0074782290    98   0.0045126113    99  -0.0011830119   101
-0.0446313639   102  -0.0560606692   103   0.2905583232   104  -0.0000000002   105
 0.1172071362   107  -0.0000000001   108   0.0038785629   110  -0.0000000005   111
-0.0082267365   113   0.0011834651   115   0.2225345571   116   0.0787777798   117
-0.0152898742   118  -0.0000000002   119
# ORBITAL         17  NAO =      90
-0.0014020326     1  -0.0068956525     2   0.0009658087     3   0.2467463560     4
-0.0150840636     5   0.1010428983     7  -0.0041186505     8  -0.0213912436    10
 0.0047152733    11   0.0207049577    13   0.0003249521    15   0.1090368977    16
 0.0565120453    17   0.0032595528    18  -0.0081550919    19  -0.0014020678    21
-0.0068956523    22   0.0009658091    23  -0.2467463563    24  -0.0150840638    25
-0.1010428983    27  -0.0041186508    28  -0.0213912436    30  -0.0047152734    31
 0.0207049580    33   0.0003249521    35   0.1090368981    36   0.0565120453    37
-0.0032595527    38  -0.0081550918    39   0.0028093586    41   0.0137482733    42
-0.0019741520    43   0.2207188322    44  -0.0000000000    45   0.0939446698    47
 0.0000000001    48  -0.0274316281    50  -0.0000000006    51   0.0288061575    53
-0.0006463540    55  -0.2181730829    56  -0.1130340303    57  -0.0108702108    58
 0.0000000003    59  -0.0014020302    61  -0.0068956520    62   0.0009658091    63
-0.2467463568    64   0.0150840636    65  -0.1010428979    67   0.0041186509    68
-0.0213912440    70   0.0047152734    71   0.0207049583    73   0.0003249522    75
 0.1090368979    76   0.0565120450    77  -0.0032595523    78   0.0081550920    79
-0.0014020296    81  -0.0068956524    82   0.0009658087    83   0.2467463565    84
 0.0150840633    85   0.1010428980    87   0.0041186504    88  -0.0213912437    90
-0.0047152733    91   0.0207049580    93   0.0003249522    95   0.1090368978    96
 0.0565120452    97   0.0032595526    98   0.0081550920    99   0.0028092657   101
 0.0137482731   102  -0.0019741520   103  -0.2207188329   104   0.0000000001   105
-0.0939446697   107   0.0000000002   108  -0.0274316280   110   0.0000000006   111
 0.0288061582   113  -0.0006463534   115  -0.2181730832   116  -0.1130340301   117
 0.0108702120   118   0.0000000003   119
# ORBITAL         18  NAO =      90
-0.0023749933     1  -0.0118884368     2   0.0017334005     3  -0.0150505701     4
 0.2293750064     5  -0.0041065876     7   0.0962994757     8  -0.0046652132    10
 0.0297409795    11   0.0034826782    13   0.0005612091    15   0.1889435721    16
 0.0978899218    17  -0.0081612949    18  -0.0061659303    19   0.0023750060    21
 0.0118884365    22  -0.0017334006    23  -0.0150505706    24  -0.2293750071    25
-0.0041065875    27  -0.0962994755    28   0.0046652135    30   0.0297409784    31
-0.0034826783    33  -0.0005612086    35  -0.1889435720    36  -0.0978899215    37
-0.0081612954    38   0.0061659300    39   0.0000000032    41   0.0000000001    42
-0.0000000001    43   0.0000000002    44   0.2554222892    45   0.0000000000    47
 0.1034266384    48   0.0000000003    50   0.0215760829    51  -0.0000000002    53
 0.0000000002    55  -0.0000000000    56  -0.0000000001    57  -0.0000000001    58
 0.0079574230    59  -0.0023750295    61  -0.0118884363    62   0.0017334008    63
 0.0150505703    64  -0.2293750078    65   0.0041065877    67  -0.0962994753    68
-0.0046652137    70   0.0297409788    71   0.0034826785    73   0.0005612087    75
 0.1889435718    76   0.0978899212    77   0.0081612957    78   0.0061659304    79
 0.0023750118    81   0.0118884367    82  -0.0017334007    83   0.0150505701    84
 0.2293750071    85   0.0041065877    87   0.0962994753    88   0.0046652134    90
 0.0297409792    91  -0.0034826785    93  -0.0005612092    95  -0.1889435718    96
-0.0978899216    97   0.0081612954    98  -0.0061659305    99   0.0000000018   101
 0.0000000003   102   0.0000000001   103   0.0000000001   104  -0.2554222907   105
 0.0000000000   107  -0.1034266382   108  -0.0000000003   110   0.0215760822   111
 0.0000000001   113  -0.0000000004   115   0.0000000000   116  -0.0000000001   117
 0.0000000000   118  -0.0079574229   119
# ORBITAL         19  NAO =       5
-0.6051361517     6  -0.5306934722     9   0.0161429500    12   0.0279606935    14
-0.0181292011    20
# ORBITAL         20  NAO =       5
-0.6051361519    26  -0.5306934721    29  -0.0161429499    32   0.0279606936    34
-0.0181292011    40
# ORBITAL         21  NAO =       5
-0.6051369321    46  -0.5306928695    49  -0.0322859902    52   0.0000000004    54
-0.0181286009    60
# ORBITAL         22  NAO =       5
-0.6051361524    66  -0.5306934715    69  -0.0161429500    72  -0.0279606933    74
-0.0181292011    80
# ORBITAL         23  NAO =       5
-0.6051361522    86  -0.5306934717    89   0.0161429499    92  -0.0279606934    94
-0.0181292011   100
# ORBITAL         24  NAO =       5
-0.6051369320   106  -0.5306928697   109   0.0322859897   112   0.0000000007   114
-0.0181286008   120
$end

2.2.4.2. Computational Results

The final energy can be found as -230.59097 hartree, and the resonance Kekulé structures is 24.5 kcal/mol, which is the stablization energy to delocalize the $\pi$ orbitals in a cycloalkene.

          ******  BOND ORDER  ******

ATOM 1        ATOM 2           DIST     BOND ORDER

 1 C           2 H            1.093       0.980
 1 C           3 C            1.399       1.425
 3 C           4 H            1.093       0.980
 3 C           5 C            1.399       1.011
 5 C           6 H            1.093       0.980
 5 C           7 C            1.399       1.425
 7 C           8 H            1.093       0.980
 7 C           9 C            1.399       1.012
 9 C          10 H            1.093       0.980
 1 C          11 C            1.399       1.011
 9 C          11 C            1.399       1.425
11 C          12 H            1.093       0.980

The bond orders show that the CC bondings are not equivalent now, there are 3 CC bondings with bond order 1.425, close to double bonding, and 3 bondings with 1.011, close to single bonding. The computational results suggest that the resonance from Kekulé structrues mainly stabilizes the molecule and makes all CC bondings equivalent. Both resonance from Kekulé structures and equalization of CC bonds thus present the aromaticity in C₆H₆.

2.3. Menshutkin Reaction NH₃ + CH₃Cl $\rightarrow$ [NH₃CH₃]⁺ + Cl^-

2.3.1. Chemical question to be addressed

The last exercise is the Menshutkin reaction NH₃ + CH₃Cl $\rightarrow$ [NH₃CH₃]⁺ + Cl^-. The reaction is exothermic in solution. In this example, the reaction barrier in gas phase will be calculated with L-VBSCF and L-BOVB. For more information of the reaction and the computations in solution, please refer to:

2.3.2. Active/inactive electrons and orbitals

The reaction involves the breaking of the CH₃-Cl bond and building of an NH₃-CH₃ bond. As such, the lone pair on NH₃ and the CH₃-Cl bond in the reactants constitute the active space, which involves 4 electrons on 3 orbitals. The remaining 32 electrons and 16 orbitals constitute the inactive space, including the core orbitals of N, C and Cl, 3 N-H bonds, 3 C-H bonds, and 3 s, 3 p_x and 3 p_y orbitals on Cl.

2.3.3. Fragmentation and important VB structures

The reaction involves 3 fragments: NH₃, CH₃ and Cl. The reactants include non-interacting NH₃ and CH₃Cl moieties while the product includes non-interacting [NH₃CH₃]⁺ and Cl^- moieties. Following figure shows the important VB structures in reactant (a) and product (b).

_images/menshutkin-scheme13.jpg — Fig. 2.3.1 Lewis structure for the reactant (a) and product (b).

2.3.4. Basis set and its subsets based on orbital symmetry

6–31G* is the basis set used in this example. The basis functions are simply separated into subsets according to the atoms. The basis functions on each atom will be treated as a subset and the VB orbitals will be built with these subsets.

2.3.5. Structures

There are 6 VB structures involved in the active space as shown in the figure. Structures S1-S3 describe the reactant, and S3-S5 describe the product. S6 describes the electron transfer between NH₃ and Cl, and will appear in the TS region.

_images/menshutkin-scheme14.jpg — Fig. 2.3.2 6 VB structures for Menshutkin Reaction.

2.3.6. Input File

The input files of L-VBSCF for reactant and TS are given below:

Reactant:

NH3CH3cl reactant
$ctrl
nmul=1 nstr=6 iprint=3 iscf=5 nao=3 nae=4
orbtyp=hao frgtyp=atom
int=libcint basis=6-31G*
guess=read
$end
$stru
1:16 17 17 18 19
1:16 17 17 18 18
1:16 17 17 19 19
1:16 18 18 17 19
1:16 19 19 17 18
1:16 18 18 19 19
$end
$orb
1*5 4 4 1*3 4*3 4*3 4 4 1
6
6
6
6
6
1 7 8 9
2 3 4 5
6
6
6
1 7 8 9
1 7 8 9
1 7 8 9
2 3 4 5
2 3 4 5
2 3 4 5
1 7 8 9
2 3 4 5
6
$end
$geo
 N      0.000000    0.000000    0.000000
 C      0.000000    0.000000   10.000000
 H     -1.029985    0.000000    9.647297
 H      0.514992    0.891993    9.647297
 H      0.514992   -0.891993    9.647297
 Cl     0.000000    0.000000   11.778400
 H      0.939678    0.000000   -0.389227
 H     -0.469839    0.813785   -0.389227
 H     -0.469839   -0.813785   -0.389227
$end

TS:

NH3CH3cl
$ctrl
nmul=1 nstr=6 iprint=3 nao=3 nae=4 iscf=5 itmax=300
guess=read orbtyp=hao frgtyp=atom
int=libcint basis=6-31G*
$end
$stru
1:16 17 17 18 19
1:16 17 17 18 18
1:16 17 17 19 19
1:16 18 18 17 19
1:16 19 19 17 18
1:16 18 18 19 19
$end
$orb
1*5 4 4 1*3 4*3 4*3 4 4 1
6
6
6
6
6
1 7 8 9
2 3 4 5
6
6
6
1 7 8 9
1 7 8 9
1 7 8 9
2 3 4 5
2 3 4 5
2 3 4 5
1 7 8 9
2 3 4 5
6
$end
$geo
N    0.0000000000   0.0000000000  -2.4403680000
C    0.0000000000   0.0000000000  -0.6327610000
H    1.0651490000   0.0000000000  -0.4753840000
H   -0.5325745000  -0.9224460928  -0.4753840000
H   -0.5325745000   0.9224460928  -0.4753840000
CL   0.0000000000   0.0000000000   1.8067450000
H   -0.9550790000   0.0000000000  -2.8031240000
H    0.4775395000   0.8271226766  -2.8031240000
H    0.4775395000  -0.8271226766  -2.8031240000
$end

The input files are almost the same except the initial guess and geometry. The initial guess of reactant is given here. The initial guess of TS can be obtained from the computational results of reactant. For L-BOVB computations, the user may simply change ISCF=5 to ISCF=2 and append BOVB in $CTRL section. The initial guess of L-BOVB computations are from the computational results of correpsonding L-VBSCF computation.

2.3.7. Computational Results

2.3.7.1. The VB structures

Following table shows the VB structures involved in the calculation. The first 16 doubly occupied orbitals are inactive ones, orbital 17 corresponds to the lone pair on NH₃, and orbitals 18 and 19 are singly occupied orbitals on CH₃ and Cl respectively. The correspondence to the structures is also listed.

Table 2.3.1 Table VB structures involved in Menshutkin reaction
VB structure	Structure
1 ***** 1:16 17 17 18 19	S1
2 ***** 1:16 17 17 18 18	S2
3 ***** 1:16 17 17 19 19	S3
4 ***** 1:16 18 18 17 19	S6
5 ***** 1:16 19 19 17 18	S4
6 ***** 1:16 18 18 19 19	S5

2.3.7.2. Energies

Following table shows the total energies (in a.u.) and reaction barrier at the L-VBSCF and L-BOVB levels. Compared with the reference value 33.0 kcal mol ^-1 (138.1 kJ mol ^-1) by Webb and Gordon, L-VBSCF overestimates the reaction barrier due to the lack of dynamic correlation, and L-BOVB significantly improves the result. Compared with the energies obtained by L-VBSCF and L-BOVB, L-BOVB decreases the energy at the TS geometry much more than in the reactants. The results show that dynamic correlation plays an important role in reaction barrier calculations.

Table 2.3.2 Total energies (in a.u.) and reaction barrier $\Delta E^\neq$ (in kcal/mol) by L-VBSCF and L-BOVB
	L-VBSCF	L-BOVB
$E^{react}_{tot}$	-555.27775	-555.29156
$E^{TS}_{tot}$	-555.21152	-555.24558
$\Delta E^\neq$	41.6	28.8

2.3.7.3. VB wavefunction and weights

Table shows the weights of VB structures in the reactant and TS geometries. It is clear that the most important contribution comes from S1 in the reactant geometry and S3 in the TS geometry. In the TS, S4 represents the “covalent” structure of product and S3 represents the major “ionic” structure in both reactant and product. Thus, S3 represents the “bond breaking” of C-Cl and S4 shows the “bond forming” of N-C. Similar to the previous examples, L-BOVB provides larger weights for the secondary ionic structures as compared with L-VBSCF weights, which indicates that dynamic correlation plays a more important role for ionic structures than covalent ones, as already explained.

Table 2.3.3 Coulson-Chirgwin weights of VB structures in reactant and TS geometries with L-VBSCF and L-BOVB
	Reactant		TS
	L-VBSCF	L-BOVB	L-VBSCF	L-BOVB
S1	0.663	0.621	0.091	0.178
S2	0.105	0.120	-0.001	0.002
S3	0.232	0.259	0.532	0.487
S4	0.000	0.000	0.364	0.286
S5	0.000	0.000	-0.004	0.010
S6	0.000	0.000	0.017	0.037

2.4. Potential Energy Surface of LiF (Optional)

2.4.1. Introduction

LiF is a simple hetero diatomic molecule with a single bond between atoms Li and F. The dissociation of LiF is always in the interest of chemists. The so-called “Harpoon effect” makes electron transfer when the distance between Li and F even larger than the sum of their van der Waals radius, indicating an electron transfer due to the crossing of diabatic states. For details, please refer to the reference.

In this optional exercise, the user should compute a series of points along the dissociation of LiF, get the energies of total wave function and each VB structure, and plot the potential energy surface. This exercise should be done by the users themselves. Only essential clues will be given. The basis set in this exercise is cc-pVDZ, and the active space is the minimal space.

2.4.2. Computations near equilibrium

The distance R_Li-F is set to 1.5 Angstrom.

2.4.2.1. Computation with all VB structures

Note

There is only a single bond in the molecule, so the minimal active space should be (2,2).
VB structures can be automatically generated as we have seen in the example of F₂.
Use keywords ORBTYP=HAO and FRGTYP=SAO with proper definition of fragmenets in $FRAG to build the orbitals.
The active orbitals should always be placed at the last in $ORB section.

2.4.2.2. Computation with all possible covalent structures

Note

The covalent VB structures can be either picked from all VB structure computation, or generated automatically.
Orbitals obtained in previous computation can be used as initial guess.

2.4.2.3. Computation for all possible Li⁺F^- ionic structures

Note

VB structures can be picked from all VB structure computation.
Initial guess can be obtained from the orbitals of previous computation.

2.4.2.4. Computation for all possible Li^-F⁺ ionic structures

Note

VB structures can be picked from all VB structure computation.
Initial guess can be obtained from the orbitals of previous computation.

2.4.3. Computations for Other distances

Repeat the above computations at R_Li-F= 1.1, 1.3, 1.7, 2.0, 2.6, 2.8, 3.0, 4.0, 7.0 Angstrom, get the energies and plot the potential energy surface. See at which distance the diabatic states cross. Following is the example of the potential energy surface.

_images/lif-pec.jpg — Fig. 2.4.1 Potential Energy Surface of LiF with cc-pVDZ

3. Citation

You need to cite the references of XMVB as following formats:

A) J. Chem. Phys. format: The ab initio Valence Bond calculations are performed with the XMVB program.

B) American Chemical Society format: The ab initio Valence Bond calculations are performed with the XMVB program.

Table 1.2.1 Supported point groups and irreducible representations, and corresponding options for keywords
SYMM Option	Point Group	IRRP Option	Irreducible Representation
CS	\(C_\textrm{s}\)	AP, APP	\(A'\), \(A''\)
CI	\(C_\textrm{i}\)	AG, AU	\(A_\textrm{g}\), \(A_\textrm{u}\)
C2	\(C_\textrm{2}\)	A, B	\(A\), \(B\)
C2V	\(C_\textrm{2v}\)	A1, A2, B1, B2	\(A_\textrm{1}\), \(A_\textrm{2}\), \(B_\textrm{1}\), \(B_\textrm{2}\)
C2H	\(C_\textrm{2h}\)	AG, AU, BG, BU	\(A_\textrm{g}\), \(A_\textrm{u}\), \(B_\textrm{g}\), \(B_\textrm{u}\)
D2	\(D_\textrm{2}\)	A1, A2, B1, B2	\(A_\textrm{1}\), \(A_\textrm{2}\), \(B_\textrm{1}\), \(B_\textrm{2}\)
D2H	\(D_\textrm{2h}\)	AG, AU, B1G, B2G, B3G, B1U, B2U, B3U	\(A_\textrm{g}\), \(A_\textrm{u}\), \(B_\textrm{1g}\), \(B_\textrm{2g}\), \(B_\textrm{3g}\), \(B_\textrm{1u}\), \(B_\textrm{2u}\), \(B_\textrm{3u}\)