4 Self-Consistent Field Ground-State Methods

4.8 Hartree-Fock and Density-Functional Perturbative Corrections

4.8.0.1 Theory

Closely related to the dual-basis approach of Section 4.7, but somewhat more general, is the Hartree-Fock perturbative correction (HFPC) developed by Deng et al..Deng:2009a, Deng:2009b An HFPC calculation consists of an iterative HF calculation in a small primary basis followed by a single Fock matrix formation, diagonalization, and energy evaluation in a larger, secondary basis. In the following, we denote a conventional HF calculation by HF/basis, and a HFPC calculation by HFPC/primary/secondary. Using a primary basis of n functions, the restricted HF matrix elements for a 2m-electron system are

Fμν=hμν+λσnPλσ[(μν|λσ)-12(μλ|νσ)] (4.56)

Solving the Roothaan-Hall equation in the primary basis results in molecular orbitals and an associated density matrix, 𝐏. In an HFPC calculation, 𝐏 is subsequently used to build a new Fock matrix, 𝐅[1], in a larger secondary basis of N functions

Fab[1]=hab+λσnPλσ[(ab|λσ)-12(aλ|bσ)] (4.57)

where λ, σ indicate primary basis functions and a, b represent secondary basis functions. Diagonalization of 𝐅[1] affords improved molecular orbitals and an associated density matrix 𝐏[1]. The HFPC energy is given by

EHFPC=abNPab[1]hab+12abcdNPab[1]Pcd[1][2(ab|cd)-(ac|bd)] (4.58)

where a, b, c and d represent secondary basis functions. This differs from the DBHF energy evaluation where 𝐏𝐏[1], rather than 𝐏[1]𝐏[1], is used. The inclusion of contributions that are quadratic in 𝐏𝐏[1] is the key reason for the fact that HFPC is more accurate than DBHF.

Unlike dual-basis HF, HFPC does not require that the small basis be a proper subset of the large basis, and is therefore able to jump between any two basis sets. Benchmark study of HFPC on a large and diverse data set of total and reaction energies demonstrate that, for a range of primary/secondary basis set combinations, the HFPC scheme can reduce the error of the primary calculation by around two orders of magnitude at a cost of about one third that of the full secondary calculation.Deng:2009a, Deng:2009b

A density-functional version of HFPC (“DFPC”)Deng:2010 seeks to combine the low cost of pure DFT calculations using small bases and grids, with the high accuracy of hybrid calculations using large bases and grids. The DFPC approach is motivated by the dual-functional method of Nakajima and HiraoNakajima:2006 and the dual-grid scheme of Tozer et al.Tozer:1995 Combining these features affords a triple perturbation: to the functional, to the grid, and to the basis set. We call this approach density-functional “triple jumping”.

4.8.0.2 Job Control

HFPC/DFPC calculations are controlled with the following $rem. HFPT turns on the HFPC/DFPC approximation. Note that HFPT_BASIS specifies the secondary basis set.

HFPT
       Activates HFPC/DFPC calculation.
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       Single-point energy only
RECOMMENDATION:
       Use Dual-Basis to capture large-basis effects at smaller basis cost. See reference for recommended basis set, functional, and grid pairings.

HFPT_BASIS
       Specifies the secondary basis in a HFPC/DFPC calculation.
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       None
RECOMMENDATION:
       See reference for recommended basis set, functional, and grid pairings.

DFPT_XC_GRID
       Specifies the secondary grid in a HFPC/DFPC calculation.
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       None
RECOMMENDATION:
       See reference for recommended basis set, functional, and grid pairings.

DFPT_EXCHANGE
       Specifies the secondary functional in a HFPC/DFPC calculation.
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       None
RECOMMENDATION:
       See reference for recommended basis set, functional, and grid pairings.

4.8.0.3 Examples

Example 4.23  Input for a HFPC single-point calculation.

$molecule
   0 1
   H
   H   1   0.75
$end

$rem
   EXCHANGE     hf
   BASIS        cc-pVDZ  ! primary basis
   HFPT_BASIS   cc-pVQZ  ! secondary basis
   PURECART     1111     ! set to purecart of the target basis
   HFPT         true
   GEN_SCFMAN   false    ! runs in the old SCF code
$end

Example 4.24  Input for a DFPC single-point calculation.

$molecule
   0 1
   H
   H   1   0.75
$end

$rem
   JOBTYPE          sp
   METHOD           blyp               ! primary functional
   DFPT_EXCHANGE    b3lyp              ! secondary functional
   DFPT_XC_GRID     00075000302        ! secondary grid
   XC_GRID          0                  ! primary grid
   HFPT_BASIS       6-311++G(3df,3pd)  ! secondary basis
   BASIS            6-311G*            ! primary basis
   PURECART         1111
   HFPT             true
   GEN_SCFMAN       false
$end