6. Slides for Advanced Exercises
Slides for advanded exercises can be downloaded here.
7. Computing of post-VBSCF methods
7.1. 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.
7.2. Computing with BOVB method
7.2.1. 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.
7.2.2. Computational Results
In this case, 3 VB structures are generated with 1 covalent structure (0th ion structure) and 2 ionic structures (1th ion structures). These structures can be expaned into 4 determinants.
Number of 0th ion structures is: 1 from 1 to 1
Number of 0th ion determinants is: 2 from 1 to 2
Number of 1th ion structures is: 2 from 2 to 3
Number of 1th ion determinants is: 2 from 3 to 4
Total number of structures is: 3
Total number of determinants is: 4
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
7.3. Computing with VBPT2 method
7.3.1. 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.
7.3.2. 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.
7.4. Computing with hc-DFVB method
7.4.1. 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.
7.4.2. 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.