VB Exercises: Basic Part
Slides
Slides for Basic VB Exercises
Slides for basic exercises can be downloaded here.
The Understanding of H2
Introduction
As a multiconfigurational quantum chemical method, a VBSCF calculation starts with the definition of the configuration space, i.e. active space (m,n), in which m active electrons arbitrarily distribute on n active orbitals. One can write totally three VB structures for hydrogen molecule by considering all singlet spin coupling patterns in the (2,2) active space. The three VB structures, namely covalent structure \(\Phi^{cov}\), and two ionic structures \(\Phi_1^{ion}\)and \(\Phi_2^{ion}\), are written as
where \(\phi_a\) and \(\phi_b\) are respectively the atomic orbitals of hydrogens a and b, \(s_{ab}\) is the orbital overlap between \(\phi_a\) and \(\phi_b\), and the bar over an orbital means a β spin orbital. Since no covalent bond is formed in the two ionic structures, each of the ionic structure is described by \(2^0=1\) determinant. The covalent structure consists one covalent bond, and hence is comprised of \(2^1=2\) determinants, namely \(|\phi_a\overline{\phi}_b|\) and \(|\overline{\phi}_a\phi_b|\). Note that neither of the determinants \(|\phi_a\overline{\phi}_b|\) nor \(|\overline{\phi}_a\phi_b|\) is the eigenfunction of \(\hat{S}^2\), and thus should be combined to form a spin-adapted VB structure. The three VB structures and their four expanded determinants are illustrated in the following figure.
The three VB structures and their four expanded determinants of H2 The VBSCF wave function of H2 molecule can thus be written as
where the coefficients of the two ionic structures are the same according to the symmetry of the H2 molecule. With cc-pvTZ basis set, the Coulson-Chirgwin weights of the three VB structures \(\Phi^{cov}\), \(\Phi_1^{ion}\) and \(\Phi_2^{ion}\) at the equilibrium bond length (R=0.75 Å) are respectively 0.82, 0.09 and 0.09, which indicates that the covalent bonding dominants the H2 molecule, but the ionic character still play a nonnegligible role for the homopolar bond. As the H-H bond stretched, the weight of the covalent structure gradually increases while the weights of the ionic structures decrease. At the dissociation limit the H-H bonding is purely covalent, which corresponds to the two neutral hydrogen atoms after bond dissociation. The following figure plots the weights of the covalent and ionic structures along the dissociation of H2 molecule. We demonstrate again the VB theory is natural multi-determinantal, even when only one covalent structure is involved in the wave function for bond breaking. The multi-determinant feature enables VB theory to correctly treat bond breaking and strongly-correlated systems.
Computation of H2 at Stationary Point
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.
The computation is proceeded at the equilibrium bond lenght of 0.75 Å, and the basis set is cc-pVDZ. For simplicity, H atoms are located in the Z axis.
Input File
Running an XMVB job requires an input file with extension name “xmi”, which is composed of several sections. The following is an simplest example xmi input file of VBSCF calculation of H2.
H2 VBSCF with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3
BASIS=CC-PVDZ
$END
$ORB
5 5
1-5
6-10
$END
$GEO
H 0.0 0.0 0.0
H 0.0 0.0 0.75
$END
The keywords in XMVB are case insensitive. The first line is the title of the job, and followed by several sections starting with $CTRL (or $STR, $ORB, $GEO, etc.) and ending with $END. The order of the sections can be arbitrary. The keywords within each section are separated by blanks and are not limited to a single line.
In an xmi file, the $CTRL and $GEO sections, which correspond to global control and molecular geometry, must be provided by the user, and other sections might be default in some cases.
In the $CTRL section, NAE and NAO are respectively the numbers of active electrons and active orbitals, and (2,2) active space is used in this case;
STR=FULL means all the 3 VB structures in the (2,2) active space are included; BASIS=CC-PVDZ means the cc-pVDZ basis set is used.
No VB method is specified in this inputfile, and the default VBSCF calculation will be performed.
The $ORB section defines the VB orbitals. The first line specifies the numbers of basis functions to expand each orbital, each number separated by blanks.
In this case (“NAO=2”), 5 5 means the two orbitals are expanded with 5 and 5 basis functions. With cc-pVDZ basis set, the basis functions of the first H atom (shown in the first line in $GEO section) are labeled as 1-5,
while those of the second H atom (the second line in $GEO section) are labeled as 6-10. Thus, the second line 1-5 in $orb section means the first orbital is expanded by the basis functions of the first H atom,
while the third line 6-10 means the second orbital is expanded by the basis functions of the second H atom.
Output File
The main XMVB output file has an “xmo” extension. The output file starts with the XMVB banner, then followed by the description of VB structures used in calculation, nuclear repulsion energy, initial orbital guess, and a repetition of the input file. For a VBSCF calculation, the information of orbital optimization (the energy and gradients) at each iteration is printed. After the end of the SCF procedure, the VBSCF wave function is printed, including the Hamiltonian and overlap matrices, VB structure and determinant coefficients, weights of VB structures, and the optimized orbitals. Finally, the charge and spin population analysis, bond order and dipole moments are printed. Following is a detailed explain of H 2 (corresponding to the input file in the previous section).
In the (2,2) active space, totally 3 structures (4 determinents) are involved in the VBSCF wave function.
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 2
2 ***** 1 1
3 ***** 2 2
The user may find from the output file that the VBSCF converged after 6 iterations and the final energy is -1.14694448 hartree.
VBSCF converged in 6 iterations
Total Energy: -1.14694448
First Excited: -0.268326
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 H-H bond should be a covalent bond.
****** COEFFICIENTS OF STRUCTURES ******
1 -0.83526891 ****** 1 2
2 -0.09985465 ****** 1 1
3 -0.09985465 ****** 2 2
.
.
.
****** WEIGHTS OF STRUCTURES ******
1 0.83388006 ****** 1 2
2 0.08305997 ****** 1 1
3 0.08305997 ****** 2 2
Finally, the bond order value (0.950) in atomic population analysis shows that H-H bond should be a single bond.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 H 2 H 0.750 0.950
Computing of F2
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.
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.
In this example, we will also investigate the nature of the F-F bonding, which is not the conventional covalent or ionic bond, but is charge-shift bond.
Electron-pair bonding is a central chemical paradigm to describe and interpret the interaction between atoms in a molecule. The concepts of covalent and ionic bonds can be dated back to the ingenious hypothesis of Lewis in 1916, and are still widely accepted and used nowadays. The covalent bonding comes from the electron spin-pairing of adjacent atoms and thus strongly stabilizes the molecule, while the ionic bonding comes from the electrostatic interaction of two fragments with opposite charges. Typical covalent bonds are usually found between H, B, C, N, O atoms, while typical ionic bonds are formed with one alkali or alkaline earth metal atom and one halogen atom. However, despite great successes of traditional electron-pair bonding in explaining molecular stability and chemical reaction mechanisms, the chemical bonding even in some simple molecules, F2 for instance, cannot be unambiguously classified into covalent or ionic.
The following figure shows the dissociation curves of the total wave function and covalent structure for H2 and F2 molecules. For H2, the potential energy curve (PEC) of covalent structure is a little higher than that of the total wave function, but the topology of the both PECs is quite close. One thus concludes that the bonding in H2 is dominated by the covalent interaction between the two H atoms. For F2, however, the PEC of covalent structure is repulsive, and much higher than that of the total wave function, and the topology of the two PECs is totally different, i.e. the two F atoms tend to dissociate due to the covalent interaction. Since both the covalent and ionic interactions are non-bonding in F2, the bond dissociation energy (BDE) of F2, which is around 38.2 kcal/mol, is totally due to the resonance of the VB structures. Based on the above-mentioned novel bonding, Shaik and Hiberty et al. proposed a new type of chemical bond, charge-shift (CS) bond, in which the RE contributes over 50% to the BDE.
The dissociation curves of the total wave function and covalent structure for (a) H2 and (b) F2 molecules.
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.
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.
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 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
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=FULLXMVB generates all VB structures automatically according to a specific active space.IPRINT=3XMVB will print most information.NAO=2andNAE=2Specify the active space with 2 active orbital and 2 active electrons respectively.ORBTYP=HAOandFRGTYP=SAOThe VB orbitals are described with fragments.BASIS=CC-PVDZThe 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, pz, dxx, dyy and dzz 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.
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
Computations with 3 Structures at Dissociation Limit
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
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.
Computations with Covalent Structure at Stationary Point
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 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ
$END
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.
Resonance in C6H6
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.
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.
Computations with 5 Covalent Structures
First, we will compute the energy with all 5 covalent structures.
Input File
The input file is shown below.
C6H6
$ctrl
str=cov nao=6 nae=6 iprint=3
orbtyp=hao frgtyp=sao
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.
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
Computations with 2 Kekulé Structures
Input File
The input file is shown below.
C6H6
$ctrl
nstr=2 nao=6 nae=6 iprint=3
orbtyp=hao frgtyp=sao
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
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# ORBITAL 4 NAO = 90
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# ORBITAL 5 NAO = 90
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# ORBITAL 6 NAO = 90
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# ORBITAL 7 NAO = 90
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# ORBITAL 8 NAO = 90
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# ORBITAL 9 NAO = 90
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# ORBITAL 10 NAO = 90
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# ORBITAL 11 NAO = 90
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# ORBITAL 12 NAO = 90
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# ORBITAL 13 NAO = 90
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# ORBITAL 14 NAO = 90
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# 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.
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.
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.
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 iprint=3
orbtyp=hao frgtyp=sao
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
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# ORBITAL 7 NAO = 90
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# ORBITAL 8 NAO = 90
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# ORBITAL 9 NAO = 90
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# ORBITAL 10 NAO = 90
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# ORBITAL 11 NAO = 90
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# ORBITAL 12 NAO = 90
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# ORBITAL 13 NAO = 90
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# ORBITAL 14 NAO = 90
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# ORBITAL 15 NAO = 90
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# ORBITAL 16 NAO = 90
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# ORBITAL 17 NAO = 90
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# ORBITAL 18 NAO = 90
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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
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.
Computing of O2
Introduction
The calculation of ground spin state of O2 has been an enduring topic in the early stages of the development of quantum chemistry. The ground state of O2 is triplet (\(^3\Sigma_g^{-}\)) from experiments. One of the most well-known “failure” of VB theory is that the ground state of O2 molecule is predicted as \(^1\Delta_g\) with VB theory. Here we clarify again that the so-called “failure” comes from the naive application of hybridization followed by the simple Lewis pairing. The ab initio VB calculation is capable of correctly describing the ground state of O2. In this section, both the triplet and singlet ground states of O2 will be computed and compared.
Computations of the singlet state
The computations are proceeded at the equilibrium bond length of 1.227 Å, and the basis set is cc-pVDZ. For simplicity, O atoms are located in the Z axis. Totally 4 active oritals (\(p_x\) and \(p_y\) orbitals of each O atom) and the 6 active electrons on them are inclued in the calculation. (Readers can also include the two \(p_z\) orbitals in the active space, but only the four mentioned \(p_{\pi}\) orbitals are included in this case)
Input File
Here shows the XMVB input file for all the structures
singlet O2 VBSCF with all structures
$CTRL
STR=FULL NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=1
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
8 8 8 8 16 3 3 3 3
# ORBITAL 1 NAO = 8
-1.0001816769 1 -0.0219071559 2 -0.0208718204 3 0.0055675373 6
0.0058885238 9 0.0012121651 10 0.0012098650 13 0.0014914363 15
# ORBITAL 2 NAO = 8
-1.0001807197 16 -0.0219222377 17 -0.0208875165 18 -0.0055748376 21
-0.0058896710 24 0.0012101317 25 0.0012115362 28 0.0014916466 30
# ORBITAL 3 NAO = 8
-0.0380292983 1 0.4956131485 2 0.5430800220 3 -0.1721288256 6
-0.1265645521 9 0.0040244173 10 0.0040243680 13 -0.0013604634 15
# ORBITAL 4 NAO = 8
-0.0380587693 16 0.4956124042 17 0.5430787036 18 0.1721299458 21
0.1265653159 24 0.0040244421 25 0.0040243828 28 -0.0013600709 30
# ORBITAL 5 NAO = 16
-0.1873055284 1 -0.0320312577 2 0.1037620245 3 -0.7575342039 6
-0.1745797408 9 0.0069166781 10 0.0069169372 13 -0.0385887223 15
-0.1873136771 16 -0.0320328992 17 0.1037594786 18 0.7575341385 21
0.1745797704 24 0.0069172963 25 0.0069171279 28 -0.0385889746 30
# ORBITAL 6 NAO = 3
0.6907525874 4 0.4550551271 7 0.0256096827 12
# ORBITAL 7 NAO = 3
0.6907525665 19 0.4550551519 22 -0.0256096585 27
# ORBITAL 8 NAO = 3
0.6907525875 5 0.4550551269 8 0.0256096869 14
# ORBITAL 9 NAO = 3
0.6907525683 20 0.4550551500 23 -0.0256096533 29
$END
The global keywords listed in $CTRL section are explained below:
STR=FULLXMVB generates all VB structures automatically according to a specific active space.IPRINT=3XMVB will print most information.NAO=4andNAE=6Specify the active space with 4 active orbital and 6 active electrons respectively.ORBTYP=HAOandFRGTYP=SAOThe VB orbitals are described with fragments.BASIS=CC-PVDZThe basis set is cc-pVDZ.NMUL=1The spin multiplicity (2S+1) is 1 (The default value).
$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, pz, dxx, dyy and dzz basis functions on atom 1, which is the first F atom in this case.
$ORB section describes VB orbitals. The first line 1*4 2 1*4 means that there are 9 orbitals in this case, each of the first 4 orbitals consisting of only 1 fragment,
the 5th orbital consisting of 2 fragments, and each of the remaining 4 orbitals 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 O atom and belongs to the \(\sigma\) space since the O-O lies on Z axis.
Also, orbital 5 with fragment 1 2 shows that the orbital describes the O-O sigma bonding orbital.
$GEO section shows the geometry of O2 molecule. Both Cartesian and internal coordinates are supported.
Computational Results
In this case, totally 10 VB structures are generated. The number of 0th ion structure is 6, and the number of 1th ion structures is 4. These structures can be expaned into 16 determinants.
Number of 0th ion structures is: 6 from 1 to 6
Number of 0th ion determinants is: 12 from 1 to 12
Number of 1th ion structures is: 4 from 7 to 10
Number of 1th ion determinants is: 4 from 13 to 16
Total number of structures is: 10
Total number of determinants is: 16
Number of Structures: 10
.
.
.
The following structures are used in calculation:
1 ***** 1:5 6 6 7 7 8 9
2 ***** 1:5 6 6 8 8 7 9
3 ***** 1:5 6 6 9 9 7 8
4 ***** 1:5 7 7 8 8 6 9
5 ***** 1:5 7 7 9 9 6 8
6 ***** 1:5 8 8 9 9 6 7
7 ***** 1:5 6 6 7 7 8 8
8 ***** 1:5 6 6 7 7 9 9
9 ***** 1:5 6 6 8 8 9 9
10 ***** 1:5 7 7 8 8 9 9
Note
In this (6,4) active space, there are at least 2 active orbitals are always doubly occupied. For instnce, in structure “1:5 6 6 7 7 8 9”, the active orbitals 6 and 7 are doubly occupied, and the electrons on orbitals 8 and 9 form a covalent bond. However, this structure belongs to “0th ion structure” instead of “2th ion structure”.
The user may find from the output file that the VBSCF converged after 43 iterations and the final energy is -149.59869269 hartree.
VBSCF converged in 43 iterations
Total Energy: -149.59869269
First Excited: -149.598693
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 structures 1 and 6 are dominant. Structure 1 means the covalent bodning between py orbitals of the two O atoms, while Structure 6 means the covalent bodning between px orbitals of the two O atoms
****** COEFFICIENTS OF STRUCTURES ******
1 0.59132818 ***** 1:5 6 6 7 7 8 9
2 -0.00000000 ***** 1:5 6 6 8 8 7 9
3 0.00000000 ***** 1:5 6 6 9 9 7 8
4 0.00000000 ***** 1:5 7 7 8 8 6 9
5 0.00000000 ***** 1:5 7 7 9 9 6 8
6 -0.59163406 ***** 1:5 8 8 9 9 6 7
7 0.14679997 ***** 1:5 6 6 7 7 8 8
8 0.14680176 ***** 1:5 6 6 7 7 9 9
9 -0.14686415 ***** 1:5 6 6 8 8 9 9
10 -0.14686585 ***** 1:5 7 7 8 8 9 9
.
.
.
****** WEIGHTS OF STRUCTURES ******
1 0.40213218 ***** 1:5 6 6 7 7 8 9
2 0.00000000 ***** 1:5 6 6 8 8 7 9
3 0.00000000 ***** 1:5 6 6 9 9 7 8
4 0.00000000 ***** 1:5 7 7 8 8 6 9
5 -0.00000000 ***** 1:5 7 7 9 9 6 8
6 0.40253540 ***** 1:5 8 8 9 9 6 7
7 0.04881265 ***** 1:5 6 6 7 7 8 8
8 0.04881349 ***** 1:5 6 6 7 7 9 9
9 0.04885274 ***** 1:5 6 6 8 8 9 9
10 0.04885354 ***** 1:5 7 7 8 8 9 9
Finally, the bond order value (1.347) in atomic population analysis shows that O-O bond should be a double bond.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 O 2 O 1.227 1.347
Computations of the triplet state
The geometry, basis set and active space are the same as the singlet calculation in the previuos section.
Input File
Here shows the XMVB input file for all the structures
triplet O2 VBSCF with all structures
$CTRL
STR=FULL NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=3
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
8 8 8 8 16 3 3 3 3
# ORBITAL 1 NAO = 8
-1.0001816769 1 -0.0219071559 2 -0.0208718204 3 0.0055675373 6
0.0058885238 9 0.0012121651 10 0.0012098650 13 0.0014914363 15
# ORBITAL 2 NAO = 8
-1.0001807197 16 -0.0219222377 17 -0.0208875165 18 -0.0055748376 21
-0.0058896710 24 0.0012101317 25 0.0012115362 28 0.0014916466 30
# ORBITAL 3 NAO = 8
-0.0380292983 1 0.4956131485 2 0.5430800220 3 -0.1721288256 6
-0.1265645521 9 0.0040244173 10 0.0040243680 13 -0.0013604634 15
# ORBITAL 4 NAO = 8
-0.0380587693 16 0.4956124042 17 0.5430787036 18 0.1721299458 21
0.1265653159 24 0.0040244421 25 0.0040243828 28 -0.0013600709 30
# ORBITAL 5 NAO = 16
-0.1873055284 1 -0.0320312577 2 0.1037620245 3 -0.7575342039 6
-0.1745797408 9 0.0069166781 10 0.0069169372 13 -0.0385887223 15
-0.1873136771 16 -0.0320328992 17 0.1037594786 18 0.7575341385 21
0.1745797704 24 0.0069172963 25 0.0069171279 28 -0.0385889746 30
# ORBITAL 6 NAO = 3
0.6907525874 4 0.4550551271 7 0.0256096827 12
# ORBITAL 7 NAO = 3
0.6907525665 19 0.4550551519 22 -0.0256096585 27
# ORBITAL 8 NAO = 3
0.6907525875 5 0.4550551269 8 0.0256096869 14
# ORBITAL 9 NAO = 3
0.6907525683 20 0.4550551500 23 -0.0256096533 29
$END
The only difference of this input file from that in the previuos section is that the spin multiplicity is set to 3 for the triplet state.
NMUL=3The spin multiplicity (2S+1) is 3.
Computational Results
In this case, totally 6 VB structures (or 6 VB determinants) are generated, and all of them are 0th ion structures.
Number of 0th ion structures is: 6 from 1 to 6
Number of 0th ion determinants is: 6 from 1 to 6
Total number of structures is: 6
Total number of determinants is: 6
Number of Structures: 6
.
.
.
The following structures are used in calculation:
1 ***** 1:5 6 6 7 7 8 9
2 ***** 1:5 6 6 8 8 7 9
3 ***** 1:5 6 6 9 9 7 8
4 ***** 1:5 7 7 8 8 6 9
5 ***** 1:5 7 7 9 9 6 8
6 ***** 1:5 8 8 9 9 6 7
Note
In this (6,4) active space, there are at least 2 active orbitals are always doubly occupied. Different from the calculation of singlet state, the remaining 2 electrons are in the same spin. Since two electrons with same spin cannot occupy on one spatial orbital, there is no “1th ion structures” in this case.
Note
For a structure with \(2S+1\neq1\), the last \(2S\) numbers means unpaired electrons with same spin instaed of covalent boinding in the singlet case. For instance, structure “1:5 6 6 7 7 8 9” in this triplet calculation means there are two unpaired electrons with same spin on orbitals 8 and 9.
The user may find from the output file that the VBSCF converged after 43 iterations and the final energy is -149.62744434 hartree.
VBSCF converged in 43 iterations
Total Energy: -149.62744434
First Excited: -149.440051
Note
Compared with the energy of singlet state (-149.59869269 hartree), we see the VB theroy correctly predict the ground state as triplet!
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 structures 3 and 4 are mainly dominant. Structure 3 means there are two unpaired electrons with same spin on px orbital of O2 and py orbital of O1. Structure 4 means there are two unpaired electrons with same spin on px orbital of O1 and py orbital of O2. The structures 2 and 5 also give nonneglegible contributions, but are less important than structures 3 and 4. This is because both structures 2 and 5 means there are two unpaired electrons with same spin on one O atom.
****** COEFFICIENTS OF STRUCTURES ******
1 -0.00000000 ***** 1:5 6 6 7 7 8 9
2 0.24299537 ***** 1:5 6 6 8 8 7 9
3 -0.55305281 ***** 1:5 6 6 9 9 7 8
4 -0.55305265 ***** 1:5 7 7 8 8 6 9
5 0.24299211 ***** 1:5 7 7 9 9 6 8
6 0.00000000 ***** 1:5 8 8 9 9 6 7
.
.
.
****** WEIGHTS OF STRUCTURES ******
1 0.00000000 ***** 1:5 6 6 7 7 8 9
2 0.12068238 ***** 1:5 6 6 8 8 7 9
3 0.37931891 ***** 1:5 6 6 9 9 7 8
4 0.37931871 ***** 1:5 7 7 8 8 6 9
5 0.12068001 ***** 1:5 7 7 9 9 6 8
6 0.00000000 ***** 1:5 8 8 9 9 6 7
Finally, the bond order value (1.779) in atomic population analysis shows that the triplet O2 is more stable than the singlet whose bond order is 1.347.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 O 2 O 1.227 1.779
Computations with selected structures
In the previous calculations of O2, the structures in the VB wave function are all automatically generated (STR=FULL). However, it’s possible to perform a VB calculation with selected structures. The structures can be selected by either chemical intuition or a previous VB calculation. Followings are input files of singlet and triplet O2 with selected structures.
Input File
Here shows the XMVB input file with selected structures for both singlet and triplet O2.
singlet O2 VBSCF with selected structures
$CTRL
NSTR=2 NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=1
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$STR
1:5 6 6 7 7 8 9
1:5 8 8 9 9 6 7
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
8 8 8 8 16 3 3 3 3
# ORBITAL 1 NAO = 8
-1.0001816769 1 -0.0219071559 2 -0.0208718204 3 0.0055675373 6
0.0058885238 9 0.0012121651 10 0.0012098650 13 0.0014914363 15
# ORBITAL 2 NAO = 8
-1.0001807197 16 -0.0219222377 17 -0.0208875165 18 -0.0055748376 21
-0.0058896710 24 0.0012101317 25 0.0012115362 28 0.0014916466 30
# ORBITAL 3 NAO = 8
-0.0380292983 1 0.4956131485 2 0.5430800220 3 -0.1721288256 6
-0.1265645521 9 0.0040244173 10 0.0040243680 13 -0.0013604634 15
# ORBITAL 4 NAO = 8
-0.0380587693 16 0.4956124042 17 0.5430787036 18 0.1721299458 21
0.1265653159 24 0.0040244421 25 0.0040243828 28 -0.0013600709 30
# ORBITAL 5 NAO = 16
-0.1873055284 1 -0.0320312577 2 0.1037620245 3 -0.7575342039 6
-0.1745797408 9 0.0069166781 10 0.0069169372 13 -0.0385887223 15
-0.1873136771 16 -0.0320328992 17 0.1037594786 18 0.7575341385 21
0.1745797704 24 0.0069172963 25 0.0069171279 28 -0.0385889746 30
# ORBITAL 6 NAO = 3
0.6907525874 4 0.4550551271 7 0.0256096827 12
# ORBITAL 7 NAO = 3
0.6907525665 19 0.4550551519 22 -0.0256096585 27
# ORBITAL 8 NAO = 3
0.6907525875 5 0.4550551269 8 0.0256096869 14
# ORBITAL 9 NAO = 3
0.6907525683 20 0.4550551500 23 -0.0256096533 29
$END
“NSTR=2” means 2 structures are involved in the VB wave function. When “NSTR=value” is specified in the input, the “$STR” section, which describes the details of the selected structures, must be provided.
triplet O2 VBSCF with selected structures
$CTRL
NSTR=2 NAO=4 NAE=6 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ NMUL=3
GUESS=READ
$END
$FRAG
1*6
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 1
PXDXZ 2
PYDYZ 1
PYDYZ 2
$END
$ORB
1*4 2 1*4
1
2
1
2
1 2
3
4
5
6
$END
$STR
1:5 6 6 9 9 7 8
1:5 7 7 8 8 6 9
$END
$GEO
O 0.0 0.0 0.0
O 0.0 0.0 1.227
$END
$GUS
8 8 8 8 16 3 3 3 3
# ORBITAL 1 NAO = 8
-1.0001816769 1 -0.0219071559 2 -0.0208718204 3 0.0055675373 6
0.0058885238 9 0.0012121651 10 0.0012098650 13 0.0014914363 15
# ORBITAL 2 NAO = 8
-1.0001807197 16 -0.0219222377 17 -0.0208875165 18 -0.0055748376 21
-0.0058896710 24 0.0012101317 25 0.0012115362 28 0.0014916466 30
# ORBITAL 3 NAO = 8
-0.0380292983 1 0.4956131485 2 0.5430800220 3 -0.1721288256 6
-0.1265645521 9 0.0040244173 10 0.0040243680 13 -0.0013604634 15
# ORBITAL 4 NAO = 8
-0.0380587693 16 0.4956124042 17 0.5430787036 18 0.1721299458 21
0.1265653159 24 0.0040244421 25 0.0040243828 28 -0.0013600709 30
# ORBITAL 5 NAO = 16
-0.1873055284 1 -0.0320312577 2 0.1037620245 3 -0.7575342039 6
-0.1745797408 9 0.0069166781 10 0.0069169372 13 -0.0385887223 15
-0.1873136771 16 -0.0320328992 17 0.1037594786 18 0.7575341385 21
0.1745797704 24 0.0069172963 25 0.0069171279 28 -0.0385889746 30
# ORBITAL 6 NAO = 3
0.6907525874 4 0.4550551271 7 0.0256096827 12
# ORBITAL 7 NAO = 3
0.6907525665 19 0.4550551519 22 -0.0256096585 27
# ORBITAL 8 NAO = 3
0.6907525875 5 0.4550551269 8 0.0256096869 14
# ORBITAL 9 NAO = 3
0.6907525683 20 0.4550551500 23 -0.0256096533 29
$END
Computational Results
The VBSCF calculations with selected structure for singlet and triplet O2 are respectivlely -149.54466456 and -149.55699157 hartree. It’s not surprising the energies with selected structures are higher than those with all structures in the actve space. The calculations with selected structures again correctly predict the ground state of O2 as triplet.
Readers can further check and compared other quantities in the output files.
VB Exercises: Advanced Part
Slides
Slides for Advanced Exercises
Slides for advanded exercises can be downloaded here.
Computing of post-VBSCF methods
Introduction
The multiconfiguration nature of VBSCF method enables VB theory to successfully describe the complicated electronic structure in strongly correlated systems. The wave function of a strongly correlated system is dominated by more than one configuration state function (CSF) or VB structure. The electronic correlation due to this degeneracy is known as static correlation. On the other hand, a more accurate description of strongly correlated systems also requires the inclusion of dynamic correlation. The dynamic correlation could be introduced by using post-SCF methods, such as perturbation theory (PT), configuration interaction (CI), coupled-cluster (CC), or density functional theory (DFT), etc. Therefore, more precise and advanced VB methods have been proposed to involve both static and dynamic correlations. The commonly used post-VBSCF methods include breathing-orbital valence bond (BOVB), valence bond configuration interaction (VBCI), valence bond perturbation theory (VBPT2), and density functional valence bond (DFVB).
The BOVB is a classical VB method which mimics a ‘breathing’ effect by allowing the orbitals to be different in different structures. For H2 molecule, the VBSCF wave function is
while the BOVB wave function is written as
\[\Psi^{BOVB}=C_1(|\phi_a\overline{\phi}_b|-|\overline{\phi}_a\phi_b|))+C_2|\phi^{'}_a\overline{\phi^{'}}_a|+C_3|\phi^{'}_b\overline{\phi^{'}}_b|\]
Therefore, BOVB method gives lower energies compared with VBSCF by introducing dynamic correlation with extra degrees of freedom of the orbital coefficients. There are various levels of BOVB, including L-BOVB, D-BOVB, SL-BOVB, and SD-BOVB. In L-BOVB, all orbitals are strictly localized (HAOs), while OEOS are used in D-BOVB. “S” in SL-BOVB and SD-BOVB stands for splitting the doubly occupied orbitals for ionic bond.
The VBCI is the most precise (though also the most expensive) classical VB method. In VBCI method, all the occupied orbitals are required to be HAOs so that the strictly localized virtual orbitals are constructed by using a projector localized on one atom or fragment. Then the CI procedure is perfomred by allowing the electron excitation from the occupied otbials to virtual ones whithin each atom or fragment.
The BOVB and VBCI are classical post-VBSCF methods, and the wave functions are constructed by HAOs. In VBPT2, however, the virtual orbitals are orthogonal by allowing the delocalization over the whole molecule. Therefore, the excited VB structures in VBPT2 are not limited to local excitation. This means the chemical pictures of excited VB structures are less clear than those in VBCI, but the computational cost of VBPT2 is much lower than VBCI.
Kohn-Sham DFT is widely applied in electronic structure calculations, due to its cheap computational cost for dynamic correlation. Various DFVB methods which combine KS-DFT and VBSCF have been proposed to efficiently include dynamic correlation in VBSCF wave function. In the dynamic correlated DFVB (dc-DFVB) method, the DFT correlation is straightforward added to the VBSCF energy. In the Hamiltonian corrected DFVB (hc-DFVB) method, the DFT correlation is added to each Hamiltonian matrix element. In a series of hybrid DFVB (λ-DFVB) methods, the electron-electron interaction is divided into the wave function and DFT parts with a hybrid parameter λ, which is related to the multiconfiguration character of the studied molecule.
In this section, three post-VBSCF methods (BOVB, VBPT2 and hc-DFVB) are applied to the computing of resonance energy of F2 molecule. All 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.
Computing with BOVB method
Input File
F2 BOVB calculation with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3 BOVB
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ GUESS=READ
$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
$VEC
8 8 8 8 3 3 3 3 8 8
# ORBITAL 1 NAO = 8
1.0008523635 1 -0.0002451908 2 -0.0033475855 3 0.0007822794 6
-0.0004308019 9 -0.0009672775 10 -0.0009672547 11 -0.0015047199 12
# ORBITAL 2 NAO = 8
1.0008523652 16 -0.0002450266 17 -0.0033475641 18 -0.0007832238 21
0.0004299611 24 -0.0009663380 25 -0.0009641029 26 -0.0015084314 27
# ORBITAL 3 NAO = 8
-0.0046583295 1 -0.5140815122 2 -0.5639080344 3 0.0635587135 6
0.0463521927 9 -0.0000387922 10 -0.0000387928 11 0.0006078983 12
# ORBITAL 4 NAO = 8
-0.0046580393 16 -0.5140815112 17 -0.5639080329 18 -0.0635586939 21
-0.0463521769 24 -0.0000388134 25 -0.0000388635 26 0.0006079803 27
# ORBITAL 5 NAO = 3
0.6920248541 4 0.4582846681 7 0.0074447939 14
# ORBITAL 6 NAO = 3
0.6920248517 19 0.4582846708 22 -0.0074447974 29
# ORBITAL 7 NAO = 3
0.6920248541 5 0.4582846681 8 0.0074447939 15
# ORBITAL 8 NAO = 3
0.6920248499 20 0.4582846729 23 -0.0074447973 30
# ORBITAL 9 NAO = 8
0.0453208934 1 -0.0554693805 2 0.0178207146 3 -0.7191076297 6
-0.4212641367 9 0.0105222178 10 0.0105222177 11 -0.0469454207 12
# ORBITAL 10 NAO = 8
0.0453208934 16 -0.0554693808 17 0.0178207148 18 0.7191076300 21
0.4212641368 24 0.0105222168 25 0.0105222108 26 -0.0469454121 27
$END
Readers can compare this input file with that of VBSCF. In the BOVB calculation, the “BOVB” keyword is specified. The “GUESS=READ” keyword means the orbital guess is taken from the $VEC (can be also written as “$GUS”) section. The optimal orbitals can be found in the file with “.orb” extension after a calculation is done.
Note
In post-VBSCF calculation, it’s always recommended to provide the optimal VBSCF orbitals as initial guess.
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
Breathing Orbitals
Structure 2: 1 -> 11
2 -> 12
3 -> 13
4 -> 14
5 -> 15
6 -> 16
7 -> 17
8 -> 18
9 -> 19
Structure 3: 1 -> 20
2 -> 21
3 -> 22
4 -> 23
5 -> 24
6 -> 25
7 -> 26
8 -> 27
10 -> 28
.
.
.
Number of Structures: 3
The following structures are used in calculation:
1 ***** 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10
2 ***** 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19
3 ***** 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28
Since the orbitals in each structure are different in BOVB wave function, the orbitals in structure 1 are labeled as 1-10 (the same as those given by the user). In structure 2, which corresponds to the ionic structure “1:8 9 9” in VBSCF, the 9 orbitals are labeled as 11-19. In structure 3, which corresponds to the ionic structure “1:8 10 10” in VBSCF, the 9 orbitals are labeled as 20-28.
The user may find from the output file that the final BOVB energy is -198.78317897 hartree, which is lower than the VBSCF energy (-198.75115493 hartree)
due to the inclusion of dynamic correlation in BOVB.
Following are the coefficients and weights of generated VB structures. Both coefficients and weights show that the covalent structure is also the dominant one in BOVB calculation. Readers can find the BOVB weight of the covalent structure (0.70862656) is smaller than VBSCF (0.77586). This phenomenon can often be found in post-VBSCF calculations since the structures with higher energies in VBSCF gain more correlation in post-VBSCF methods.
****** COEFFICIENTS OF STRUCTURES ******
1 -0.75348845 ****** 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10
2 -0.26504301 ****** 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19
3 -0.26504287 ****** 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28
.
.
.
****** WEIGHTS OF STRUCTURES ******
1 0.70862656 ****** 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10
2 0.14568688 ****** 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19
3 0.14568656 ****** 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28
Computing with VBPT2 method
Input File
F2 VBPT2 calculation with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3 VBPT2
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ GUESS=READ
$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
$VEC
8 8 8 8 3 3 3 3 8 8
# ORBITAL 1 NAO = 8
1.0008523635 1 -0.0002451908 2 -0.0033475855 3 0.0007822794 6
-0.0004308019 9 -0.0009672775 10 -0.0009672547 11 -0.0015047199 12
# ORBITAL 2 NAO = 8
1.0008523652 16 -0.0002450266 17 -0.0033475641 18 -0.0007832238 21
0.0004299611 24 -0.0009663380 25 -0.0009641029 26 -0.0015084314 27
# ORBITAL 3 NAO = 8
-0.0046583295 1 -0.5140815122 2 -0.5639080344 3 0.0635587135 6
0.0463521927 9 -0.0000387922 10 -0.0000387928 11 0.0006078983 12
# ORBITAL 4 NAO = 8
-0.0046580393 16 -0.5140815112 17 -0.5639080329 18 -0.0635586939 21
-0.0463521769 24 -0.0000388134 25 -0.0000388635 26 0.0006079803 27
# ORBITAL 5 NAO = 3
0.6920248541 4 0.4582846681 7 0.0074447939 14
# ORBITAL 6 NAO = 3
0.6920248517 19 0.4582846708 22 -0.0074447974 29
# ORBITAL 7 NAO = 3
0.6920248541 5 0.4582846681 8 0.0074447939 15
# ORBITAL 8 NAO = 3
0.6920248499 20 0.4582846729 23 -0.0074447973 30
# ORBITAL 9 NAO = 8
0.0453208934 1 -0.0554693805 2 0.0178207146 3 -0.7191076297 6
-0.4212641367 9 0.0105222178 10 0.0105222177 11 -0.0469454207 12
# ORBITAL 10 NAO = 8
0.0453208934 16 -0.0554693808 17 0.0178207148 18 0.7191076300 21
0.4212641368 24 0.0105222168 25 0.0105222108 26 -0.0469454121 27
$END
Readers can compare this input file with that of VBSCF. In the VBPT2 calculation, the “VBPT2” keyword is specified. The orbital guess is taken from the $VEC section.
Computational Results
In VBPT2 calculation, the reference is the VBSCF wav function, and therefore aslo includes 3 structures or 4 determinants.
The user may find from the output file that the final VBPT2 energy is -199.09559220 hartree.
Note
Since the virtual orbitals in VBPT2 are delocazlied, the chemical pictures of exciteted structures are less clear than those in VBCI, and the structural weights are not computed in VBPT2.
Computing with hc-DFVB method
Input File
F2 hc-DFVB calculation with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3 HC-DFVB=B3LYP
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ GUESS=READ
$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
$VEC
8 8 8 8 3 3 3 3 8 8
# ORBITAL 1 NAO = 8
1.0008523635 1 -0.0002451908 2 -0.0033475855 3 0.0007822794 6
-0.0004308019 9 -0.0009672775 10 -0.0009672547 11 -0.0015047199 12
# ORBITAL 2 NAO = 8
1.0008523652 16 -0.0002450266 17 -0.0033475641 18 -0.0007832238 21
0.0004299611 24 -0.0009663380 25 -0.0009641029 26 -0.0015084314 27
# ORBITAL 3 NAO = 8
-0.0046583295 1 -0.5140815122 2 -0.5639080344 3 0.0635587135 6
0.0463521927 9 -0.0000387922 10 -0.0000387928 11 0.0006078983 12
# ORBITAL 4 NAO = 8
-0.0046580393 16 -0.5140815112 17 -0.5639080329 18 -0.0635586939 21
-0.0463521769 24 -0.0000388134 25 -0.0000388635 26 0.0006079803 27
# ORBITAL 5 NAO = 3
0.6920248541 4 0.4582846681 7 0.0074447939 14
# ORBITAL 6 NAO = 3
0.6920248517 19 0.4582846708 22 -0.0074447974 29
# ORBITAL 7 NAO = 3
0.6920248541 5 0.4582846681 8 0.0074447939 15
# ORBITAL 8 NAO = 3
0.6920248499 20 0.4582846729 23 -0.0074447973 30
# ORBITAL 9 NAO = 8
0.0453208934 1 -0.0554693805 2 0.0178207146 3 -0.7191076297 6
-0.4212641367 9 0.0105222178 10 0.0105222177 11 -0.0469454207 12
# ORBITAL 10 NAO = 8
0.0453208934 16 -0.0554693808 17 0.0178207148 18 0.7191076300 21
0.4212641368 24 0.0105222168 25 0.0105222108 26 -0.0469454121 27
$END
Readers can compare this input file with that of VBSCF, and find that the only difference is the “HC-DFVB=B3LYP” keyword is specified. This keyword is specified for a hc-DFVB calculation, and B3LYP functional is used in this case. Currently, the program supports only LDA, GGA and their hybrid functionals; meta-GGA and corresponding hybrid functional are not avaiable temporaily.
Computational Results
In hc-DFVB calculation, the reference is the VBSCF wave function, and therefore aslo includes 3 structures or 4 determinants.
The user may find from the output file that the final hc-DFVB energy is -199.40139652 hartree. The hc-DFVB correlation energy (-0.65024078 hartree) is also printed out.
Readers can find in the output file the “MATRIX OF EC HAMILTONIAN FOR HC-DFVB”, which is a correlation correction matrix in the structure basis. In this case, it reads as
****** MATRIX OF EC HAMILTONIAN FOR HC-DFVB ******
1 2 3
1 -0.647844 -0.240498 -0.240498
2 -0.240498 -0.658429 -0.047883
3 -0.240498 -0.047883 -0.658429
It can be found that the correlation energy of the covalent structure is -0.647844 hartree, which is smaller (in absolute value) than that of ionic structure (-0.658429 hartree).
This phenomenon, which is also found in the previous BOVB calculation, again shows that structures with higher energies in VBSCF gain more correlation in post-VBSCF methods.
Computing of diabatic states with VB theory
Introduction
Molecular dynamical process involving excited states is usually simulated under the Born-Oppenheimer adiabatic approximation, in which the adiabatic electronic states are used to construct the potential energy surfaces (PESs). Adiabatic states are eigenstates of electronic Hamiltonian, and thus unique for a given quantum chemical method. For instance, the eigenstates obtained by solving the generalized secular equation \(\textbf{HC}=\textbf{MCE}\) in VBSCF are adiabatic states. In nonadiabatic dynamics, to solve the quantum equation of motion requires the evaluation of nuclear-momentum couplings (NMCs) \(\textbf{F}_{KL}\)
where \(\Psi_K(\textbf{r};\textbf{R})\) is the electronic wave function of state K with parametrical dependence on the nuclear coordinates \(\textbf{R}\). Specially, \(\textbf{F}_{KL}\) in adiabatic state basis is called nonadiabatic couplings (NACs). The complexity arises near the conical intersections where the NACs have singularities due to the rapid change of the character of the adiabatic wave function. The sudden change of the adiabatic wave function can be understood according to the following figure, which could be interpreted as a conical intersection between one covalent state and one ionic state. Suppose that the adiabatic state \(\Psi_1\) is mainly covalent and \(\Psi_2\) is mainly ionic before the avoided crossing point (\(R^c\)). Then singularity of \(\textbf{F}_{12}\) occurs at \(R^c\) when \(\Psi_1\) rapidly changes to ionic and \(\Psi_2\) rapidly changes to covalent. It is possible to remove such singularity by constructing the PESs with diabatic states. In diabatic state basis, the electronic Hamiltonian is not diagonal, and the effect of the NMCs should be negligible compared with the off-diagonal elements of the electronic Hamiltonian. Therefore, the definition of diabatic states is not unique.
Illustration of adiabatic states (solid lines) and diabatic states (dashed lines).
Various schemes based on molecular orbital theory have been proposed to construct the diabatic states. On the contrary, the construction of diabatic states with VB theory seems more straightforward, since each VB structure exactly corresponds to a Lewis structure, and thus never changes its character along the PES. In this section, two ways to construct diabtic states with VB theory are presented for LiF molecule.
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 exercise, users will compute a series of points along the dissociation of LiF for the first two electronic states, get the energies of both the adiabatic and diabatic states with VB theory, and plot the potential energy surface. The basis set in this exercise is cc-pVDZ, and the active space is the minimal (2,2) space, which includes the \(2s\) orbital of Li atom and \(2p_z\) orbital of F atom.
Computations of adiabatic states
The distance RLi-F is 1.5 Å in the example input file below.
Input file
The first two adiabatic states of LiF
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ
WSTATE(1)=0.5,0.5 GUESS=READ
$END
$FRAG
1*4
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 2
PYDYZ 2
$END
$ORB
1*7
1
2
2
3
4
1
2
$END
$GEO
Li 0.0 0.0 0.0
F 0.0 0.0 1.5
$END
$VEC
8 8 8 3 3 8 8
# ORBITAL 1 NAO = 8
1.0006124741 1 0.0010947827 2 -0.0001697524 3 -0.0054823903 6
0.0022720913 9 -0.0024855501 10 -0.0024855494 11 -0.0029391815 12
# ORBITAL 2 NAO = 8
-0.9878720649 16 0.0820885570 17 0.1032423966 18 -0.0041896343 21
-0.0056696719 24 0.0001113470 25 0.0001115264 26 0.0012300179 27
# ORBITAL 3 NAO = 8
0.1608417532 16 0.4836528017 17 0.5898820135 18 -0.0253431901 21
-0.0323439728 24 -0.0057810288 25 -0.0057799058 26 -0.0006485649 27
# ORBITAL 4 NAO = 3
-0.6610782115 19 -0.4931501640 22 0.0078682530 29
# ORBITAL 5 NAO = 3
-0.6610781961 20 -0.4931501514 23 0.0078713335 30
# ORBITAL 6 NAO = 8
-0.0485005246 1 0.3835061386 2 0.5812431299 3 -0.3104635019 6
-0.0450765604 9 0.0269723639 10 0.0269723820 11 -0.0250362698 12
# ORBITAL 7 NAO = 8
-0.0016023849 16 -0.0212026440 17 0.0287929283 18 -0.6406201457 21
-0.5152310378 24 -0.0077601696 25 -0.0077603299 26 0.0077058038 27
$END
Note
There is only a single bond in the molecule, so the minimal active space should be (2,2).
VB structures can be automatically generated as we have seen in the example of F2.
Use keywords
ORBTYP=HAOandFRGTYP=SAOwith proper definition of fragmenets in$FRAGto build the orbitals.The active orbitals should always be placed at the last in
$ORBsection.Orbitals obtained in previous computation can be used as initial guess.
The keyword “WSTATE(1)=0.5,0.5” in the $CTRL section is specified for the state-averaged VBSCF calulation. The state-averaged scheme is needed for the calculation of excited states in multiconfigurational methods. In state-averaged scheme, the wave function for each electronic state is optimized variationally to minimize the averaged energy (E) of interested N states as
where \(E_i\) is the energy of state i, and \(w_i\) is the corresponding weight in the averaged scheme. In most cases, the electronc states are averaged with same weights. In the “WSTATE(K)= \(w_1,w_2,...\)" keyword, the averaged scheme is described as following: the Kth state is averaged with weight w1, the \((K+1)\)th state is averaged with weight w2, and so on. In this case, the first two electronic states are averaged, and the weight of the first state (ground state) is 0.5, and the weight of the second state (first excited state) is also 0.5.
The “GUESS=READ” keyword means the orbital guess is taken from the $VEC section. For a calculation of potential energy curve, the orbital guess is often chosen as the optimal orbitals of the molecuar geometry nearby. The optimal orbitals can be found in the file with “.orb” extension after a calculation is done.
Computation of diabatic states with chemical insights
For LiF molecule, the definition of the first two diabatic states is straightforward, i.e. the covalent structure Li-F and the ionic structure Li+ F-. The input files for the calculations of the two diabatic states are shown below.
The covalent diabatic state of LiF
$CTRL
NSTR=1 NAO=2 NAE=2 IPRINT=3
ORBTYP=HAO FRGTYP=SAO GUESS=READ
BASIS=CC-PVDZ
$END
$FRAG
1*4
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 2
PYDYZ 2
$END
$ORB
1*7
1
2
2
3
4
1
2
$END
$STR
1:5 6 7
$END
$GEO
Li 0.0 0.0 0.0
F 0.0 0.0 1.5
$END
$VEC
8 8 8 3 3 8 8
# ORBITAL 1 NAO = 8
-1.0011678861 1 -0.0022121270 2 0.0023212935 3 -0.0025335037 6
0.0016875131 9 0.0025270489 10 0.0023274646 13 0.0026480262 15
# ORBITAL 2 NAO = 8
-1.0008168299 16 0.0036845277 17 0.0071487300 18 -0.0001661450 21
-0.0003147690 24 0.0010390175 25 0.0010419561 28 0.0015319075 30
# ORBITAL 3 NAO = 8
-0.0101097268 16 -0.5128227302 17 -0.5703498101 18 0.0266052512 21
0.0288144250 24 0.0021878838 25 0.0021691970 28 -0.0016764977 30
# ORBITAL 4 NAO = 3
0.6907944105 19 0.4597105881 22 -0.0059097276 27
# ORBITAL 5 NAO = 3
0.6907940010 20 0.4597105227 23 -0.0059816036 29
# ORBITAL 6 NAO = 8
-0.0238842027 1 -0.4984202033 2 -0.5284694265 3 0.2687769028 6
0.1065714103 9 -0.0107075041 10 -0.0107111179 13 0.0798765082 15
# ORBITAL 7 NAO = 8
0.0080197261 16 -0.0318421895 17 -0.0744506221 18 0.7073285005 21
0.4345585123 24 0.0040544825 25 0.0040620001 28 -0.0046080448 30
$END
The ionic diabatic state of LiF
$CTRL
NSTR=1 NAO=2 NAE=2 IPRINT=3
ORBTYP=HAO FRGTYP=SAO GUESS=READ
BASIS=CC-PVDZ
$END
$FRAG
1*4
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 2
PYDYZ 2
$END
$ORB
1*7
1
2
2
3
4
1
2
$END
$STR
1:5 7 7
$END
$GEO
Li 0.0 0.0 0.0
F 0.0 0.0 1.5
$END
$VEC
8 8 8 3 3 8 8
# ORBITAL 1 NAO = 8
-1.0004203054 1 0.0002038534 2 0.0005481657 3 0.0132303641 6
-0.0065003814 9 0.0021658077 10 0.0021658077 13 0.0027685384 15
# ORBITAL 2 NAO = 8
-1.0008305986 16 -0.0017058286 17 0.0010077161 18 0.0003735826 21
-0.0001046315 24 0.0011858422 25 0.0011849757 28 0.0011520210 30
# ORBITAL 3 NAO = 8
0.0205934347 16 -0.4655906963 17 -0.6286682097 18 0.0355102258 21
0.0460224079 24 0.0093958692 25 0.0093958715 28 0.0024192082 30
# ORBITAL 4 NAO = 3
0.6265556203 19 0.5304582609 22 -0.0098651162 27
# ORBITAL 5 NAO = 3
0.6265556195 20 0.5304582617 23 -0.0098651162 29
# ORBITAL 6 NAO = 8
-0.1494486185 1 -0.7392177847 2 0.2171735656 3 -0.7734875826 6
0.2332816045 9 0.3272572338 10 0.3272572338 13 -0.5469570507 15
# ORBITAL 7 NAO = 8
0.0096938330 16 0.0583635389 17 0.0122710208 18 0.6211959886 21
0.5329337539 24 0.0088600826 25 0.0088600766 28 -0.0093045652 30
$END
For the covalent diabatic state Li-F, the corrsponding structure is “1:5 6 7”; For the ionic diabatic state Li+ F-, the corrsponding structure is “1:5 7 7”. From the output files, readers can find the energy of the covalent structure is -106.74408537 hartree, and the energy of the ionic structure is -106.89954237 hartree. Therefore, the bonding of Li-F is dominated by the ionic structure near equilibrium geometry. After bond dissociation, however, the bonding of Li-F is dominated by the covalent structure. The potential energy curves of these two diabatic states are plotted in the following figure.
The potential energy curves of two diabatic states obtained by chmical insights.
Note
Careful readers may notice the potential energy curves of VBSCF adiabatic states are not consistent with those in reference. This is because VBSCF is not accurate enough, and better curves can be obtained with post-VBSCF methods.
Computation of diabatic states with VBCAD
Instead of diabatization scheme with chemical insights, the valence-bond-based compression approach for diabatization (VBCAD), however, is a black-box-like diabatization scheme. In VBCAD, the electronic Hamiltonian is first compressed into a low dimensional matrix Hpre-dia containing only the interested electronic states, and then the diabatic Hamiltonian (Hdia) is obtained by a transformation to Hpre-dia. The transformation is determined by maximizing the separation of VB structures in different electronic states. For more details, please refer to reference.
An example input file for VBCAD calculation is shown below
The first two diabatic states of LiF with VBCAD
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3
ORBTYP=HAO FRGTYP=SAO
BASIS=CC-PVDZ
WSTATE(1)=0.5,0.5 GUESS=READ VBCAD
$END
$FRAG
1*4
SPZDXXDYYDZZ 1
SPZDXXDYYDZZ 2
PXDXZ 2
PYDYZ 2
$END
$ORB
1*7
1
2
2
3
4
1
2
$END
$GEO
Li 0.0 0.0 0.0
F 0.0 0.0 1.5
$END
$VEC
8 8 8 3 3 8 8
# ORBITAL 1 NAO = 8
1.0006124741 1 0.0010947827 2 -0.0001697524 3 -0.0054823903 6
0.0022720913 9 -0.0024855501 10 -0.0024855494 11 -0.0029391815 12
# ORBITAL 2 NAO = 8
-0.9878720649 16 0.0820885570 17 0.1032423966 18 -0.0041896343 21
-0.0056696719 24 0.0001113470 25 0.0001115264 26 0.0012300179 27
# ORBITAL 3 NAO = 8
0.1608417532 16 0.4836528017 17 0.5898820135 18 -0.0253431901 21
-0.0323439728 24 -0.0057810288 25 -0.0057799058 26 -0.0006485649 27
# ORBITAL 4 NAO = 3
-0.6610782115 19 -0.4931501640 22 0.0078682530 29
# ORBITAL 5 NAO = 3
-0.6610781961 20 -0.4931501514 23 0.0078713335 30
# ORBITAL 6 NAO = 8
-0.0485005246 1 0.3835061386 2 0.5812431299 3 -0.3104635019 6
-0.0450765604 9 0.0269723639 10 0.0269723820 11 -0.0250362698 12
# ORBITAL 7 NAO = 8
-0.0016023849 16 -0.0212026440 17 0.0287929283 18 -0.6406201457 21
-0.5152310378 24 -0.0077601696 25 -0.0077603299 26 0.0077058038 27
$END
The keyword “VBCAD” is specified for a black-box-like diabatization with VB theory. Currently, VBCAD supports diabatization with only 2 electronic states.
In the output file, the Hamiltonian in the diabatic state basis is printed out as
****** Diabatic Hamiltonian ******
1 2
1 -106.714613 0.013448
2 0.013448 -106.877770
followed by the printing of energy and wave function of each diabatic state.
VBCAD DIABATIC STATE: 1
VBCAD DIABATIC ENERGY: -106.714613
****** COEFFICIENTS OF STRUCTURES ******
1 1.00169 ****** 1:5 6 7
2 -0.01716 ****** 1:5 6 6
3 -0.02505 ****** 1:5 7 7
CC Weights
1 1.00123 ****** 1:5 6 7
2 -0.00058 ****** 1:5 6 6
3 -0.00065 ****** 1:5 7 7
.
.
.
VBCAD DIABATIC STATE: 2
VBCAD DIABATIC ENERGY: -106.877770
****** COEFFICIENTS OF STRUCTURES ******
1 0.02599 ****** 1:5 6 7
2 0.00044 ****** 1:5 6 6
3 -1.00099 ****** 1:5 7 7
CC Weights
1 -0.00065 ****** 1:5 6 7
2 0.00000 ****** 1:5 6 6
3 1.00065 ****** 1:5 7 7
It can be found from the CC weights section, the covalent structure Li-F dominats the ground diabatic state, while the ionic structure Li+ F- dominats the first excited diabatic state. This is in consistent with the fact that near equilibrium Li-F bond length, the ground diabatic state is described as covalent, and the first excited diabatic state is described as ionic.
The potential energy curves of these two VBCAD states together with the adiabatic states are plotted in the following figure.
The potential energy curves of the VBCAD states together with the adiabatic states.
Menshutkin Reaction NH3 + CH3Cl \(\rightarrow\) [NH3CH3]+ + Cl-
Chemical question to be addressed
This 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:
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 px and 3 py orbitals on Cl.
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).
Lewis structure for the reactant (a) and product (b).
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.
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.
6 VB structures for Menshutkin Reaction.
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 nao=3 nae=4
orbtyp=hao frgtyp=atom
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 itmax=300
guess=read orbtyp=hao frgtyp=atom
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 append BOVB in $CTRL section. The initial guess of L-BOVB computations are from the computational results of correpsonding L-VBSCF computation.
Computational Results
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.
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 |
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.
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 |
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 structures with higher VBSCF energies, as already explained.
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 |
Charge-shift bonding in propellane
Introduction
In this section, the charge-shift bonding nature of the C-C bridge bond (shown in the following figue) in the [1.1.1]propellane compound (propellane for short in this section) is investigated. As mentioned in Section 3.2.1, the charge-shift bond is identified if the resonance energy of the bond exceeds 50% of the bonding energy.
The structure of the propellane compound.
The computations are proceeded with the given geometry in the following input files, and the basis set is cc-pVDZ. The calulation is at BOVB level.
Computations of the resonance energy
propellane BOVB calculation with 3 structures
$CTRL
STR=FULL NAO=2 NAE=2 IPRINT=3 BOVB
ORBTYP=HAO GUESS=READ
BASIS=CC-PVDZ
$END
$ORB
1*5 2*12 1*2
1 # 1 C1 1s
2 # 2 C2 1s
3 # 3 C3 1s
4 # 4 C4 1s
5 # 5 C5 1s
1 3 # 6 C1-C3 sigma
2 3 # 7 C2-C3 sigma
1 4 # 8 C1-C4 sigma
2 4 # 9 C2-C4 sigma
1 5 # 10 C1-C5 sigma
2 5 # 11 C2-C5 sigma
3 6 # 12 C3-H6 sigma
3 7 # 13 C3-H7 sigma
4 8 # 14 C4-H8 sigma
4 9 # 15 C4-H9 sigma
5 10 # 16 C5-H10 sigma
5 11 # 17 C5-H11 sigma
1 # 18 C1-C2 sigma C1
2 # 19 C1-C2 sigma C2
$END
$GEO
C -0.80339320 -0.00000002 -0.00026581
C 0.80339315 0.00000002 -0.00026580
C -0.00000000 0.00000000 1.30328491
C 0.00000001 1.12889613 -0.65192077
C -0.00000002 -1.12889613 -0.65192076
H 0.00000002 0.90072681 1.88197724
H 0.00000001 -0.90072681 1.88197723
H -0.00000001 2.07996500 -0.16030252
H 0.00000003 1.18070087 -1.72126710
H 0.00000001 -1.18070087 -1.72126709
H 0.00000000 -2.07996501 -0.16030253
$END
$VEC
15 15 15 15 15 30 30 30 30 30 30 20 20 20 20 20 20 15 15
# ORBITAL 1 NAO = 15
1.0013044300 1 -0.0148185006 2 -0.0270614351 3 0.0014157398 4
-0.0000001541 5 0.0000015008 6 -0.0017992610 7 -0.0000067568 8
-0.0000032993 9 -0.0008838473 10 -0.0000009796 11 0.0000002515 12
-0.0015986173 13 0.0000013752 14 -0.0016010424 15
# ORBITAL 2 NAO = 15
1.0013053868 16 -0.0148178712 17 -0.0270673561 18 -0.0014157125 19
-0.0000002118 20 0.0000016740 21 0.0018009244 22 -0.0000010308 23
0.0000054004 24 -0.0008824051 25 -0.0000002306 26 -0.0000009051 27
-0.0015985372 28 0.0000006106 29 -0.0015993992 30
# ORBITAL 3 NAO = 15
0.9991206001 31 -0.0171860691 32 -0.0162161271 33 0.0000007618 34
0.0000013055 35 -0.0006957167 36 -0.0000014569 37 0.0000001571 38
0.0000790563 39 -0.0020670891 40 -0.0000011719 41 0.0000010731 42
-0.0021065954 43 -0.0000022108 44 -0.0023469285 45
# ORBITAL 4 NAO = 15
0.9991202122 46 -0.0171853822 47 -0.0162125157 48 0.0000006933 49
-0.0005994343 50 0.0003494060 51 -0.0000008239 52 0.0000652009 53
-0.0000372870 54 -0.0020683845 55 -0.0000009837 56 -0.0000004101 57
-0.0022864381 58 0.0001199393 59 -0.0021660489 60
# ORBITAL 5 NAO = 15
0.9991212097 61 -0.0171850657 62 -0.0162182626 63 0.0000004025 64
0.0006008934 65 0.0003484130 66 0.0000035522 67 -0.0000702075 68
-0.0000394260 69 -0.0020683512 70 0.0000006972 71 0.0000002999 72
-0.0022870879 73 -0.0001190917 74 -0.0021658981 75
# ORBITAL 6 NAO = 30
-0.0177774926 1 -0.2162705930 2 -0.2298798504 3 0.0706361562 4
0.0000024378 5 -0.3238519858 6 0.0667063348 7 -0.0000256996 8
-0.1556546218 9 0.0097518542 10 -0.0000027624 11 -0.0301554752 12
0.0092026397 13 -0.0000002305 14 -0.0263744821 15 -0.0147709460 31
-0.1460413984 32 -0.1155006162 33 0.2650092251 34 0.0000006955 35
0.1839524192 36 0.1404511764 37 0.0000407801 38 0.0925295733 39
-0.0107900693 40 0.0000018292 41 -0.0282003496 42 0.0136549918 43
0.0000054096 44 -0.0097832753 45
# ORBITAL 7 NAO = 30
-0.0177658023 16 -0.2162486406 17 -0.2299525549 18 -0.0706350937 19
-0.0000014106 20 -0.3238462405 21 -0.0666827544 22 -0.0000053780 23
-0.1555802125 24 0.0097646308 25 -0.0000018502 26 0.0301631205 27
0.0092087511 28 0.0000029753 29 -0.0263731317 30 -0.0147687986 31
-0.1460413510 32 -0.1155156453 33 -0.2649967418 34 -0.0000072210 35
0.1839401239 36 -0.1405170026 37 -0.0000348214 38 0.0924884616 39
-0.0107989165 40 -0.0000016736 41 0.0282085984 42 0.0136617248 43
-0.0000053124 44 -0.0097795494 45
# ORBITAL 8 NAO = 30
-0.0177577308 1 -0.2162013994 2 -0.2299673947 3 0.0706110630 4
-0.2804620260 5 0.1619371044 6 0.0666333857 7 -0.1348756187 8
0.0778258361 9 0.0097734916 10 -0.0261267575 11 0.0150869471 12
-0.0174618406 13 0.0178097102 14 0.0003162100 15 -0.0147732236 46
-0.1460578616 47 -0.1155220923 48 0.2650144529 49 0.1592704233 50
-0.0920250702 51 0.1404068731 52 0.0800667609 53 -0.0462473103 54
-0.0107879126 55 -0.0244083850 56 0.0141015632 57 -0.0039034180 58
0.0117339554 59 0.0077884189 60
# ORBITAL 9 NAO = 30
0.0177761911 16 0.2162548982 17 0.2298552881 18 0.0706285957 19
0.2805096055 20 -0.1619483728 21 0.0667282475 22 0.1348077090 23
-0.0778328435 24 -0.0097796081 25 -0.0261254688 26 0.0150935495 27
0.0174656722 28 -0.0178011546 29 -0.0003043329 30 0.0147931236 46
0.1461082080 47 0.1153584115 48 0.2650336900 49 -0.1592773444 50
0.0920130590 51 0.1405345297 52 -0.0800770669 53 0.0462541047 54
0.0108069299 55 -0.0244257552 56 0.0141067392 57 0.0039234196 58
-0.0117232129 59 -0.0077872362 60
# ORBITAL 10 NAO = 30
0.0177805057 1 0.2162405637 2 0.2298076785 3 -0.0706044220 4
-0.2804965339 5 -0.1619531390 6 -0.0666833559 7 -0.1347393815 8
-0.0777019246 9 -0.0097445984 10 -0.0261176194 11 -0.0150718170 12
0.0174729106 13 0.0177945974 14 -0.0002864287 15 0.0147874057 61
0.1460888402 62 0.1154517089 63 -0.2650232945 64 0.1592757818 65
0.0920142609 66 -0.1405860261 67 0.0801337784 68 0.0462885854 69
0.0108070870 70 -0.0244331942 71 -0.0141107441 72 0.0039410936 73
0.0117131673 74 -0.0078047684 75
# ORBITAL 11 NAO = 30
0.0177799561 16 0.2162227894 17 0.2298148247 18 0.0705868963 19
-0.2804588970 20 -0.1619115261 21 0.0666441698 22 -0.1348267513 23
-0.0778975430 24 -0.0097385288 25 0.0261260325 26 0.0150938317 27
0.0174863028 28 0.0178098786 29 -0.0003106634 30 0.0147619518 61
0.1460362911 62 0.1156613683 63 0.2650100271 64 0.1592697337 65
0.0920215396 66 0.1403723237 67 0.0801289455 68 0.0462892649 69
0.0107835415 70 0.0244067802 71 0.0140992591 72 0.0039118995 73
0.0117295466 74 -0.0077996863 75
# ORBITAL 12 NAO = 20
0.0172973458 31 0.2021529787 32 0.1246606235 33 -0.0000031809 34
0.3042972540 35 0.1970108564 36 0.0000068028 37 0.1300804271 38
0.0908986389 39 -0.0079455903 40 0.0000048018 41 -0.0000039238 42
0.0117793218 43 0.0139996205 44 -0.0073463152 45 0.4270402124 76
0.1295468232 77 -0.0000053093 78 -0.0294631198 79 -0.0195166579 80
# ORBITAL 13 NAO = 20
-0.0173282009 31 -0.2022108198 32 -0.1243909531 33 -0.0000096109 34
0.3043243458 35 -0.1970216211 36 -0.0000042424 37 0.1300749867 38
-0.0909331248 39 0.0079484887 40 -0.0000056809 41 0.0000028882 42
-0.0117996319 43 0.0139959547 44 0.0073369181 45 -0.4270900640 81
-0.1296000887 82 0.0000006580 83 -0.0294618950 84 0.0195172636 85
# ORBITAL 14 NAO = 20
0.0173002408 46 0.2021118790 47 0.1246240723 48 0.0000022713 49
0.3227889720 50 0.1649529052 51 -0.0000148620 52 0.1438492365 53
0.0672219489 54 -0.0079367406 55 0.0000175697 56 -0.0000089690 57
0.0079591066 58 0.0165643938 59 -0.0034869406 60 0.4270093758 86
0.1295884488 87 0.0000016180 88 -0.0316194875 89 -0.0157531870 90
# ORBITAL 15 NAO = 20
-0.0173086687 46 -0.2021939367 47 -0.1245611168 48 0.0000003610 49
-0.0183875471 50 0.3620460821 51 0.0000673331 52 -0.0136415635 53
0.1580963305 54 0.0079527536 55 -0.0000113874 56 0.0000032002 57
0.0130569871 58 -0.0025806853 59 -0.0175049037 60 -0.4270744827 91
-0.1295400125 92 0.0000001650 93 0.0021637053 94 -0.0352738044 95
# ORBITAL 16 NAO = 20
-0.0173183346 61 -0.2022026360 62 -0.1245109940 63 0.0000075545 64
0.0183988865 65 0.3620424314 66 -0.0000395096 67 0.0136658395 68
0.1581128420 69 0.0079480868 70 -0.0000049906 71 -0.0000080201 72
0.0130454991 73 0.0025855278 74 -0.0175242163 75 -0.4270686231 96
-0.1295457954 97 -0.0000005864 98 -0.0021655869 99 -0.0352696895 100
# ORBITAL 17 NAO = 20
0.0173197873 61 0.2021511454 62 0.1244587778 63 -0.0000011094 64
-0.3228115619 65 0.1649560698 66 0.0000052783 67 -0.1438497907 68
0.0671949240 69 -0.0079405965 70 0.0000104455 71 -0.0000034930 72
0.0079482684 73 -0.0165642389 74 -0.0034671716 75 0.4270459111 101
0.1296314976 102 -0.0000009897 103 0.0316266447 104 -0.0157561697 105
# ORBITAL 18 NAO = 15
-0.0163776922 1 -0.0583204663 2 0.1324850019 3 -0.6739726665 4
-0.0000444708 5 -0.0000310706 6 -0.4551573242 7 -0.0000875557 8
-0.0000314392 9 -0.0756705415 10 -0.0000033820 11 -0.0000057832 12
0.0323892303 13 0.0000190463 14 0.0323603700 15
# ORBITAL 19 NAO = 15
-0.0163754625 16 -0.0583193461 17 0.1324498525 18 0.6739423059 19
0.0000015891 20 -0.0000003256 21 0.4551921058 22 -0.0000787438 23
-0.0000153122 24 -0.0756649095 25 0.0000030203 26 -0.0000219594 27
0.0323852880 28 -0.0000064669 29 0.0323956416 30
$END
Careful readers may wonder why the $FRAG section is missing in this input file. This is because, each atom is defined as a fragment in default, if $FRAG section is not specified by the user. In the default case, atom 1 is defined as fragment 1, atom 2 is defined as fragment 2, and so on. The description of each orbital in the calculation is shown in the $ORB section. In the input file, the contents after “#” in a line is regarded as comments, and will not be read by the program. The “GUESS=READ” keyword is specified for the BOVB calculation, and the VBSCF optimal orbitals are provided as initial guess in the $VEC section.
To obtain the resonance energy, another input file to comput the energy with one covalent structure is needed. The BOVB energy of the total 3 structures is -192.68236940 hartree,
and that of one covalent structure is -192.58034082 hartree. Therefore, the resonance enenrgy of the C-C bridge bond is 64.02 kcal/mol at BOVB level.
Note
Careful readers may notice the BOVB energy with one covalent structure is exactly identical to the VBSCF enenrgy. This is because BOVB includes dynamic corrlation by allowing different orbitals in different structures. When the wave funtion contains only one structure, the BOVB and VBSCF wave functions are identical.
Computations of the bonding energy
The bodning energy is the energy difference between the equilibrium geometry and the geometry after bond breaking. In the previous F2 moleucle, the bond breaking geometry is simply described by setting the F-F distance as a large value (10 Å for instance). In this propellane case, however, it’s impossible to stretch the C-C bridge bond while keeping the molecular skeleton.
To approximately estimated the bonding energy of the C-C bridge bond, we define a quasiclassical (QC) state as a reference non-bonded state. For a single bond, the bonded state is the covalent state \(\Phi^{cov}=\frac{1}{\sqrt{2+2s_{ab}}}(|\phi_a\overline{\phi}_b|-|\overline{\phi}_a\phi_b|)\), and the QC state can be either \(|\phi_a\overline{\phi}_b|\) or \(|\overline{\phi}_a\phi_b|\). Each of these determinants has only quasi-classical interactions between the two fragments and is therefore non-bonding, while the covalent bonding arises from the interaction of the two determinants that give rise to the singlet-coupled covalent state \(\Phi^{cov}\). This covalent bond energy, complemented by the resonance energy stabilization due to the mixing of the ionic structures into the covalent structure measures the “in-situ” bonding energy. As such, the “in-situ” bonidng energy, which is the energy difference between the QC state and state with totally 3 structures provides an approximated measure of the C-C bridge boning energy.
The following is a input file to compuete the QC state.
QC state calculation of propellane
$CTRL
WFNTYP=DET NSTR=1 NAO=2 NAE=2 IPRINT=3
ORBTYP=HAO GUESS=READ
BASIS=CC-PVDZ
$END
$STR
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19
$END
$ORB
1*5 2*12 1*2
1 # 1 C1 1s
2 # 2 C2 1s
3 # 3 C3 1s
4 # 4 C4 1s
5 # 5 C5 1s
1 3 # 6 C1-C3 sigma
2 3 # 7 C2-C3 sigma
1 4 # 8 C1-C4 sigma
2 4 # 9 C2-C4 sigma
1 5 # 10 C1-C5 sigma
2 5 # 11 C2-C5 sigma
3 6 # 12 C3-H6 sigma
3 7 # 13 C3-H7 sigma
4 8 # 14 C4-H8 sigma
4 9 # 15 C4-H9 sigma
5 10 # 16 C5-H10 sigma
5 11 # 17 C5-H11 sigma
1 # 18 C1-C2 sigma C1
2 # 19 C1-C2 sigma C2
$END
$GEO
C -0.80339320 -0.00000002 -0.00026581
C 0.80339315 0.00000002 -0.00026580
C -0.00000000 0.00000000 1.30328491
C 0.00000001 1.12889613 -0.65192077
C -0.00000002 -1.12889613 -0.65192076
H 0.00000002 0.90072681 1.88197724
H 0.00000001 -0.90072681 1.88197723
H -0.00000001 2.07996500 -0.16030252
H 0.00000003 1.18070087 -1.72126710
H 0.00000001 -1.18070087 -1.72126709
H 0.00000000 -2.07996501 -0.16030253
$END
$VEC
15 15 15 15 15 30 30 30 30 30 30 20 20 20 20 20 20 15 15
# ORBITAL 1 NAO = 15
1.0013044300 1 -0.0148185006 2 -0.0270614351 3 0.0014157398 4
-0.0000001541 5 0.0000015008 6 -0.0017992610 7 -0.0000067568 8
-0.0000032993 9 -0.0008838473 10 -0.0000009796 11 0.0000002515 12
-0.0015986173 13 0.0000013752 14 -0.0016010424 15
# ORBITAL 2 NAO = 15
1.0013053868 16 -0.0148178712 17 -0.0270673561 18 -0.0014157125 19
-0.0000002118 20 0.0000016740 21 0.0018009244 22 -0.0000010308 23
0.0000054004 24 -0.0008824051 25 -0.0000002306 26 -0.0000009051 27
-0.0015985372 28 0.0000006106 29 -0.0015993992 30
# ORBITAL 3 NAO = 15
0.9991206001 31 -0.0171860691 32 -0.0162161271 33 0.0000007618 34
0.0000013055 35 -0.0006957167 36 -0.0000014569 37 0.0000001571 38
0.0000790563 39 -0.0020670891 40 -0.0000011719 41 0.0000010731 42
-0.0021065954 43 -0.0000022108 44 -0.0023469285 45
# ORBITAL 4 NAO = 15
0.9991202122 46 -0.0171853822 47 -0.0162125157 48 0.0000006933 49
-0.0005994343 50 0.0003494060 51 -0.0000008239 52 0.0000652009 53
-0.0000372870 54 -0.0020683845 55 -0.0000009837 56 -0.0000004101 57
-0.0022864381 58 0.0001199393 59 -0.0021660489 60
# ORBITAL 5 NAO = 15
0.9991212097 61 -0.0171850657 62 -0.0162182626 63 0.0000004025 64
0.0006008934 65 0.0003484130 66 0.0000035522 67 -0.0000702075 68
-0.0000394260 69 -0.0020683512 70 0.0000006972 71 0.0000002999 72
-0.0022870879 73 -0.0001190917 74 -0.0021658981 75
# ORBITAL 6 NAO = 30
-0.0177774926 1 -0.2162705930 2 -0.2298798504 3 0.0706361562 4
0.0000024378 5 -0.3238519858 6 0.0667063348 7 -0.0000256996 8
-0.1556546218 9 0.0097518542 10 -0.0000027624 11 -0.0301554752 12
0.0092026397 13 -0.0000002305 14 -0.0263744821 15 -0.0147709460 31
-0.1460413984 32 -0.1155006162 33 0.2650092251 34 0.0000006955 35
0.1839524192 36 0.1404511764 37 0.0000407801 38 0.0925295733 39
-0.0107900693 40 0.0000018292 41 -0.0282003496 42 0.0136549918 43
0.0000054096 44 -0.0097832753 45
# ORBITAL 7 NAO = 30
-0.0177658023 16 -0.2162486406 17 -0.2299525549 18 -0.0706350937 19
-0.0000014106 20 -0.3238462405 21 -0.0666827544 22 -0.0000053780 23
-0.1555802125 24 0.0097646308 25 -0.0000018502 26 0.0301631205 27
0.0092087511 28 0.0000029753 29 -0.0263731317 30 -0.0147687986 31
-0.1460413510 32 -0.1155156453 33 -0.2649967418 34 -0.0000072210 35
0.1839401239 36 -0.1405170026 37 -0.0000348214 38 0.0924884616 39
-0.0107989165 40 -0.0000016736 41 0.0282085984 42 0.0136617248 43
-0.0000053124 44 -0.0097795494 45
# ORBITAL 8 NAO = 30
-0.0177577308 1 -0.2162013994 2 -0.2299673947 3 0.0706110630 4
-0.2804620260 5 0.1619371044 6 0.0666333857 7 -0.1348756187 8
0.0778258361 9 0.0097734916 10 -0.0261267575 11 0.0150869471 12
-0.0174618406 13 0.0178097102 14 0.0003162100 15 -0.0147732236 46
-0.1460578616 47 -0.1155220923 48 0.2650144529 49 0.1592704233 50
-0.0920250702 51 0.1404068731 52 0.0800667609 53 -0.0462473103 54
-0.0107879126 55 -0.0244083850 56 0.0141015632 57 -0.0039034180 58
0.0117339554 59 0.0077884189 60
# ORBITAL 9 NAO = 30
0.0177761911 16 0.2162548982 17 0.2298552881 18 0.0706285957 19
0.2805096055 20 -0.1619483728 21 0.0667282475 22 0.1348077090 23
-0.0778328435 24 -0.0097796081 25 -0.0261254688 26 0.0150935495 27
0.0174656722 28 -0.0178011546 29 -0.0003043329 30 0.0147931236 46
0.1461082080 47 0.1153584115 48 0.2650336900 49 -0.1592773444 50
0.0920130590 51 0.1405345297 52 -0.0800770669 53 0.0462541047 54
0.0108069299 55 -0.0244257552 56 0.0141067392 57 0.0039234196 58
-0.0117232129 59 -0.0077872362 60
# ORBITAL 10 NAO = 30
0.0177805057 1 0.2162405637 2 0.2298076785 3 -0.0706044220 4
-0.2804965339 5 -0.1619531390 6 -0.0666833559 7 -0.1347393815 8
-0.0777019246 9 -0.0097445984 10 -0.0261176194 11 -0.0150718170 12
0.0174729106 13 0.0177945974 14 -0.0002864287 15 0.0147874057 61
0.1460888402 62 0.1154517089 63 -0.2650232945 64 0.1592757818 65
0.0920142609 66 -0.1405860261 67 0.0801337784 68 0.0462885854 69
0.0108070870 70 -0.0244331942 71 -0.0141107441 72 0.0039410936 73
0.0117131673 74 -0.0078047684 75
# ORBITAL 11 NAO = 30
0.0177799561 16 0.2162227894 17 0.2298148247 18 0.0705868963 19
-0.2804588970 20 -0.1619115261 21 0.0666441698 22 -0.1348267513 23
-0.0778975430 24 -0.0097385288 25 0.0261260325 26 0.0150938317 27
0.0174863028 28 0.0178098786 29 -0.0003106634 30 0.0147619518 61
0.1460362911 62 0.1156613683 63 0.2650100271 64 0.1592697337 65
0.0920215396 66 0.1403723237 67 0.0801289455 68 0.0462892649 69
0.0107835415 70 0.0244067802 71 0.0140992591 72 0.0039118995 73
0.0117295466 74 -0.0077996863 75
# ORBITAL 12 NAO = 20
0.0172973458 31 0.2021529787 32 0.1246606235 33 -0.0000031809 34
0.3042972540 35 0.1970108564 36 0.0000068028 37 0.1300804271 38
0.0908986389 39 -0.0079455903 40 0.0000048018 41 -0.0000039238 42
0.0117793218 43 0.0139996205 44 -0.0073463152 45 0.4270402124 76
0.1295468232 77 -0.0000053093 78 -0.0294631198 79 -0.0195166579 80
# ORBITAL 13 NAO = 20
-0.0173282009 31 -0.2022108198 32 -0.1243909531 33 -0.0000096109 34
0.3043243458 35 -0.1970216211 36 -0.0000042424 37 0.1300749867 38
-0.0909331248 39 0.0079484887 40 -0.0000056809 41 0.0000028882 42
-0.0117996319 43 0.0139959547 44 0.0073369181 45 -0.4270900640 81
-0.1296000887 82 0.0000006580 83 -0.0294618950 84 0.0195172636 85
# ORBITAL 14 NAO = 20
0.0173002408 46 0.2021118790 47 0.1246240723 48 0.0000022713 49
0.3227889720 50 0.1649529052 51 -0.0000148620 52 0.1438492365 53
0.0672219489 54 -0.0079367406 55 0.0000175697 56 -0.0000089690 57
0.0079591066 58 0.0165643938 59 -0.0034869406 60 0.4270093758 86
0.1295884488 87 0.0000016180 88 -0.0316194875 89 -0.0157531870 90
# ORBITAL 15 NAO = 20
-0.0173086687 46 -0.2021939367 47 -0.1245611168 48 0.0000003610 49
-0.0183875471 50 0.3620460821 51 0.0000673331 52 -0.0136415635 53
0.1580963305 54 0.0079527536 55 -0.0000113874 56 0.0000032002 57
0.0130569871 58 -0.0025806853 59 -0.0175049037 60 -0.4270744827 91
-0.1295400125 92 0.0000001650 93 0.0021637053 94 -0.0352738044 95
# ORBITAL 16 NAO = 20
-0.0173183346 61 -0.2022026360 62 -0.1245109940 63 0.0000075545 64
0.0183988865 65 0.3620424314 66 -0.0000395096 67 0.0136658395 68
0.1581128420 69 0.0079480868 70 -0.0000049906 71 -0.0000080201 72
0.0130454991 73 0.0025855278 74 -0.0175242163 75 -0.4270686231 96
-0.1295457954 97 -0.0000005864 98 -0.0021655869 99 -0.0352696895 100
# ORBITAL 17 NAO = 20
0.0173197873 61 0.2021511454 62 0.1244587778 63 -0.0000011094 64
-0.3228115619 65 0.1649560698 66 0.0000052783 67 -0.1438497907 68
0.0671949240 69 -0.0079405965 70 0.0000104455 71 -0.0000034930 72
0.0079482684 73 -0.0165642389 74 -0.0034671716 75 0.4270459111 101
0.1296314976 102 -0.0000009897 103 0.0316266447 104 -0.0157561697 105
# ORBITAL 18 NAO = 15
-0.0163776922 1 -0.0583204663 2 0.1324850019 3 -0.6739726665 4
-0.0000444708 5 -0.0000310706 6 -0.4551573242 7 -0.0000875557 8
-0.0000314392 9 -0.0756705415 10 -0.0000033820 11 -0.0000057832 12
0.0323892303 13 0.0000190463 14 0.0323603700 15
# ORBITAL 19 NAO = 15
-0.0163754625 16 -0.0583193461 17 0.1324498525 18 0.6739423059 19
0.0000015891 20 -0.0000003256 21 0.4551921058 22 -0.0000787438 23
-0.0000153122 24 -0.0756649095 25 0.0000030203 26 -0.0000219594 27
0.0323852880 28 -0.0000064669 29 0.0323956416 30
$END
In this input file, the “WFNTYP=DET” keyword means the wave function will be expressed in the basis of determinants. In this case, the wave function consists only one determinant, so we specify “NSTR=1” keyword. This determinant should be provided in $STR section. The format of writing a determinant is different from a structure. To write a determinant, we first write all the occupied \(\alpha\) orbitals (“1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18” in this case), followed by all the occupied \(\beta\) orbitals (“1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19” in this case).
Note
In this case, the wave funtion contains only one determinant, and therefore the BOVB and VBSCF wave functions are identical.
The energy of QC state is -192.49514835 hartree, compared with the energy of totally 3 structures (-192.68236940 hartree),
the bonding energy of the C-C bridge bond is 117.5 kcal/mol. Therefore, one can find the resonance energy of the C-C bond
is 54 % of the bonding energy, and thus conclude the C-C bridge bond in the propellane compound is a charge-shift bond.