Description

This method keyword requests a perfect-pairing General Valence Bond (GVB-PP) calculation. GVB requires one parameter: the number of perfect-pairing pairs to split; for example: GVB(4). This parameter may also be specified with the NPair option. The natural orbitals for the GVB pairs are taken from occupied and virtual orbitals of the initial guess determinant (described in the Input tab).

Input

Normally, most of the difficult input for a GVB-PP calculation involves specifying the initial guess (Link 401). This often includes alteration of orbitals to ensure the correct identification of high-spin, perfect-pairing, and closed-shell orbitals and possible reduction of SCF symmetry to account for the localized orbitals which usually represent the lowest energy solution for GVB-PP.

The GVB program reads the number of orbitals in each GVB pair (in format 40I2). The number of lines read is fixed (and normally 1), so no terminating blank line is needed. For a molecule having spin multiplicity S, N GVB pairs, and n_{1}, …, n_{N} orbitals in each pair, orbitals from the initial guess are used in the following manner by the GVB program:

- The S-1 highest occupied orbitals in the initial guess, which would have been singly occupied in an ROHF calculation, become high-spin orbitals.
- The next lower N occupied orbitals, which would have been doubly occupied in an ROHF calculation, become the first natural orbitals of the GVB pairs.
- Any remaining orbitals occupied in the guess stay closed-shell.
- The lowest n
_{1}-1 virtual orbitals become natural orbitals 2 through n_{1}of the first GVB pair, then the next n_{2}-1 orbitals are assigned to pair 2, and so on. The GVB-PP scheme does not allow an orbital to be shared by more than one GVB pair. - Any remaining (virtual) orbitals from the initial guess become virtual orbitals in the GVB calculation.

Generally Guess=Alter is required to ensure that guess-occupied orbitals, which will be used as first natural orbitals, match up with the correct guess virtual orbitals that will become the corresponding higher natural orbitals. Often it is helpful to start off with Guess=(Local,Only), examine the orbitals to determine alteration requirements, then do Guess=(Local,Alter) and GVB(NPair=N,Freeze) to allow the higher natural orbitals to become more appropriate. Finally, the full calculation can be run with Guess=Read and all orbitals optimized in the GVB. If there is any confusion or concern with the orbitals breaking symmetry, the calculation should be done with Symm=NoSCF and initially with Guess=Local. In fact, this approach is generally recommended except for those very expert users.

If the number of orbitals in a pair is negative, the root of the CI to use for that pair and the pair’s initial GVB coefficients are read in format (I2,5D15.8). This is useful if a ^{1}Σ or ^{1}Δ state is being represented as a GVB pair of the form x^{2} ± y^{2}.

Options

#### NPair

Gives the number of perfect-pairing pairs. GVB(N) is equivalent to GVB(NPair=N). NPair=0 is acceptable and results in a closed-shell or spin-restricted SCF calculation.

#### InHam=N

Read in N Hamiltonians (Fock operators, sets of coupling coefficients). This option may be combined with perfect-pairing pairs. Each Hamiltonian is read using the following syntax (format in parentheses):

NO # of orbitals in current Hamiltonian (I5) Fj Occup. # (1.0=closed-shell) (D15.8) (AJ(I), I =1,NHam) J coefficients (5D15.8) (AK(I), I =1,NHam) K coefficients (5D15.8)

Combining several orbitals with the same AJ and AK coefficients into one “shell” is not currently supported, so NO is always 1. The ham506 utility can be used to generate averaged Hamiltonians for the common case of spherical averaging in atomic calculations. The Hamiltonian coefficients are described in Bobrowicz and Goddard [Bobrowicz77]. A good introduction to the qualitative interpretation of GVB wavefunctions can be found in the review article by Goddard and Harding [Goddard78].

#### OSS

Do a two-electron, two-orthogonal-orbital open-shell singlet. This option may be combined with perfect-pairing pairs. OpenShellSinglet is a synonym for OSS.

#### Freeze

Freeze closed-shell and open-shell orbitals and first natural orbitals of GVB pairs, allowing only 2^{nd} and higher orbitals to vary. This option is useful for starting off difficult wavefunctions.

Availability

Energies, analytic gradients, and numerical frequencies.

Examples

Here is a GVB(3/6) calculation performed on singlet methylene:

# GVB(3)/6-31G(d) Pop=Full | |

Guess=(Local,LowSym,Alter) | |

GVB(3) on CH2 | |

molecule specification | |

1 4 0 2 3 9 | Guess=LowSym input |

2,3 | Guess=Alter input |

2 2 2 | GVB input |

Each of the 3 valence electron pairs is split into a GVB pair. A preliminary Guess=Only calculation was performed to determine the localized orbitals and which alterations would be required.

The perfect pairing GVB method includes the effects of intra-pair correlation but not those of inter-pair correlation. Consequently, GVB electrons pairs tend to be localized. In the case of singlet methylene, the carbon lone pair is localized even at the Hartree-Fock level. The canonical Hartree-Fock orbitals for the C-H bonds are delocalized into linear combinations (C-H_{1} + C-H_{2}) and (C-H_{1} – C-H_{2}) having A_{1} and B_{2} symmetry, respectively. In order to allow the localization in the guess to produce separate bond pairs, these two irreducible representations must be combined. Similarly, the GVB calculation itself must be told not to impose the full molecular symmetry on the orbitals, which would force them to be delocalized. Combining the A_{1} and B_{2} representations and combining the A_{2} and B_{1} representations causes the calculation to impose only C_{s} symmetry on the individual orbitals, allowing separate GVB pairs for each bond. Since the resulting pairs for each bond will be equivalent, the resulting overall wavefunction and density will still have C_{2v} symmetry.

The Guess=LowSymm keyword specifies that the irreducible representations of the molecular point group will be combined in the symmetry information used in a GVB calculation. It takes a single line of input consisting of giving the numbers of the irreducible representations to combine, where the numbers correspond to the order in which the representations are listed in the output file (they appear just after the standard orientation). For example, here is the output for a molecule with C_{2v} symmetry:

There are 4 symmetry adapted basis functions of A1 symmetry. There are 0 symmetry adapted basis functions of A2 symmetry. There are 1 symmetry adapted basis functions of B1 symmetry. There are 2 symmetry adapted basis functions of B2 symmetry.

Thus for C_{2v} symmetry, the order is A1, A2, B1, B2, referred to in the Guess=LowSym input as 1 through 4, respectively. A zero separates groups of representations to be combined, and a nine ends the list. Thus, to combine A1 with B2 and A2 with B1, thereby lowering the SCF symmetry to C_{s}, the appropriate input line is:

1 4 0 2 3 9

Since this information always requires exactly one line, no blank line terminates this section.

The order of orbitals generated after localization by the initial guess in the first job step was C-1s C-H_{1} C-H_{2} C-2s for the occupied orbitals and C-2p C-H_{1}* C-H_{2}* for the lowest virtual orbitals. Hence if no orbitals are interchanged, the C-2s lone pair would be correctly paired with the unoccupied p-orbital, but then the next lower occupied, C-H_{2}, would be paired with the next higher virtual, C-H_{1}*. So either the two bond occupied orbitals or the two bond virtual orbitals must be exchanged to match up the orbitals properly.

Finally, the one line of input to the GVB code indicates that there are 2 natural orbitals in each of the 3 GVB pairs.

This method keyword requests a perfect-pairing General Valence Bond (GVB-PP) calculation. GVB requires one parameter: the number of perfect-pairing pairs to split; for example: GVB(4). This parameter may also be specified with the NPair option. The natural orbitals for the GVB pairs are taken from occupied and virtual orbitals of the initial guess determinant (described in the Input tab).

Normally, most of the difficult input for a GVB-PP calculation involves specifying the initial guess (Link 401). This often includes alteration of orbitals to ensure the correct identification of high-spin, perfect-pairing, and closed-shell orbitals and possible reduction of SCF symmetry to account for the localized orbitals which usually represent the lowest energy solution for GVB-PP.

The GVB program reads the number of orbitals in each GVB pair (in format 40I2). The number of lines read is fixed (and normally 1), so no terminating blank line is needed. For a molecule having spin multiplicity S, N GVB pairs, and n_{1}, …, n_{N} orbitals in each pair, orbitals from the initial guess are used in the following manner by the GVB program:

- The S-1 highest occupied orbitals in the initial guess, which would have been singly occupied in an ROHF calculation, become high-spin orbitals.
- The next lower N occupied orbitals, which would have been doubly occupied in an ROHF calculation, become the first natural orbitals of the GVB pairs.
- Any remaining orbitals occupied in the guess stay closed-shell.
- The lowest n
_{1}-1 virtual orbitals become natural orbitals 2 through n_{1}of the first GVB pair, then the next n_{2}-1 orbitals are assigned to pair 2, and so on. The GVB-PP scheme does not allow an orbital to be shared by more than one GVB pair. - Any remaining (virtual) orbitals from the initial guess become virtual orbitals in the GVB calculation.

Generally Guess=Alter is required to ensure that guess-occupied orbitals, which will be used as first natural orbitals, match up with the correct guess virtual orbitals that will become the corresponding higher natural orbitals. Often it is helpful to start off with Guess=(Local,Only), examine the orbitals to determine alteration requirements, then do Guess=(Local,Alter) and GVB(NPair=N,Freeze) to allow the higher natural orbitals to become more appropriate. Finally, the full calculation can be run with Guess=Read and all orbitals optimized in the GVB. If there is any confusion or concern with the orbitals breaking symmetry, the calculation should be done with Symm=NoSCF and initially with Guess=Local. In fact, this approach is generally recommended except for those very expert users.

If the number of orbitals in a pair is negative, the root of the CI to use for that pair and the pair's initial GVB coefficients are read in format (I2,5D15.8). This is useful if a ^{1}Σ or ^{1}Δ state is being represented as a GVB pair of the form x^{2} ± y^{2}.

#### NPair

Gives the number of perfect-pairing pairs. GVB(N) is equivalent to GVB(NPair=N). NPair=0 is acceptable and results in a closed-shell or spin-restricted SCF calculation.

#### InHam=N

Read in N Hamiltonians (Fock operators, sets of coupling coefficients). This option may be combined with perfect-pairing pairs. Each Hamiltonian is read using the following syntax (format in parentheses):

NO # of orbitals in current Hamiltonian (I5) Fj Occup. # (1.0=closed-shell) (D15.8) (AJ(I), I =1,NHam) J coefficients (5D15.8) (AK(I), I =1,NHam) K coefficients (5D15.8)

Combining several orbitals with the same AJ and AK coefficients into one “shell” is not currently supported, so NO is always 1. The ham506 utility can be used to generate averaged Hamiltonians for the common case of spherical averaging in atomic calculations. The Hamiltonian coefficients are described in Bobrowicz and Goddard [Bobrowicz77]. A good introduction to the qualitative interpretation of GVB wavefunctions can be found in the review article by Goddard and Harding [Goddard78].

#### OSS

Do a two-electron, two-orthogonal-orbital open-shell singlet. This option may be combined with perfect-pairing pairs. OpenShellSinglet is a synonym for OSS.

#### Freeze

Freeze closed-shell and open-shell orbitals and first natural orbitals of GVB pairs, allowing only 2^{nd} and higher orbitals to vary. This option is useful for starting off difficult wavefunctions.

Energies, analytic gradients, and numerical frequencies.

Here is a GVB(3/6) calculation performed on singlet methylene:

# GVB(3)/6-31G(d) Pop=Full | |

Guess=(Local,LowSym,Alter) | |

GVB(3) on CH2 | |

molecule specification | |

1 4 0 2 3 9 | Guess=LowSym input |

2,3 | Guess=Alter input |

2 2 2 | GVB input |

Each of the 3 valence electron pairs is split into a GVB pair. A preliminary Guess=Only calculation was performed to determine the localized orbitals and which alterations would be required.

The perfect pairing GVB method includes the effects of intra-pair correlation but not those of inter-pair correlation. Consequently, GVB electrons pairs tend to be localized. In the case of singlet methylene, the carbon lone pair is localized even at the Hartree-Fock level. The canonical Hartree-Fock orbitals for the C-H bonds are delocalized into linear combinations (C-H_{1} + C-H_{2}) and (C-H_{1} - C-H_{2}) having A_{1} and B_{2} symmetry, respectively. In order to allow the localization in the guess to produce separate bond pairs, these two irreducible representations must be combined. Similarly, the GVB calculation itself must be told not to impose the full molecular symmetry on the orbitals, which would force them to be delocalized. Combining the A_{1} and B_{2} representations and combining the A_{2} and B_{1} representations causes the calculation to impose only C_{s} symmetry on the individual orbitals, allowing separate GVB pairs for each bond. Since the resulting pairs for each bond will be equivalent, the resulting overall wavefunction and density will still have C_{2v} symmetry.

The Guess=LowSymm keyword specifies that the irreducible representations of the molecular point group will be combined in the symmetry information used in a GVB calculation. It takes a single line of input consisting of giving the numbers of the irreducible representations to combine, where the numbers correspond to the order in which the representations are listed in the output file (they appear just after the standard orientation). For example, here is the output for a molecule with C_{2v} symmetry:

There are 4 symmetry adapted basis functions of A1 symmetry. There are 0 symmetry adapted basis functions of A2 symmetry. There are 1 symmetry adapted basis functions of B1 symmetry. There are 2 symmetry adapted basis functions of B2 symmetry.

Thus for C_{2v} symmetry, the order is A1, A2, B1, B2, referred to in the Guess=LowSym input as 1 through 4, respectively. A zero separates groups of representations to be combined, and a nine ends the list. Thus, to combine A1 with B2 and A2 with B1, thereby lowering the SCF symmetry to C_{s}, the appropriate input line is:

1 4 0 2 3 9

Since this information always requires exactly one line, no blank line terminates this section.

The order of orbitals generated after localization by the initial guess in the first job step was C-1s C-H_{1} C-H_{2} C-2s for the occupied orbitals and C-2p C-H_{1}* C-H_{2}* for the lowest virtual orbitals. Hence if no orbitals are interchanged, the C-2s lone pair would be correctly paired with the unoccupied p-orbital, but then the next lower occupied, C-H_{2}, would be paired with the next higher virtual, C-H_{1}*. So either the two bond occupied orbitals or the two bond virtual orbitals must be exchanged to match up the orbitals properly.

Finally, the one line of input to the GVB code indicates that there are 2 natural orbitals in each of the 3 GVB pairs.

Last updated on: 05 January 2017. [G16 Rev. C.01]