14 INTEGRAL-DIRECT CALCULATIONS (GDIRECT)

References:

Direct methods, general: M. Schütz, R. Lindh, and H.-J. Werner, Mol. Phys. 96, 719 (1999).
Linear scaling LMP2: M. Schütz, G. Hetzer, and H.-J. Werner J. Chem. Phys. 111, 5691 (1999).

All methods implemented in MOLPRO apart from full CI (FCI) and perturbative triple excitations (T) can be performed integral-direct, i.e., the methods are integral driven with the two-electron integrals in the AO basis being recomputed whenever needed, avoiding the bottleneck of storing these quantities on disk. For small molecules, this requires significantly more CPU time, but reduces the disk space requirements when using large basis sets. However, due to efficient prescreening techniques, the scaling of the computational cost with molecular size is lower in integral-direct mode than in conventional mode, and therefore integral-direct calculations for extended molecules may even be less expensive than conventional ones. The break-even point depends strongly on the size of the molecule, the hardware, and the basis set. Depending on the available disk space, calculations with more than 150-200 basis functions in one symmetry should normally be done in integral-direct mode.

Integral-direct calculations are requested by the DIRECT or GDIRECT directives. If one of these cards is given outside the input of specific programs it acts globally, i.e. all subsequent calculations are performed in integral-direct mode. On the other hand, if the DIRECT card is part of the input of specific programs (e.g. HF, CCSD), it affects only this program. The GDIRECT directive is not recognized by individual programs and always acts globally. Normally, all calculations in one job will be done integral-direct, and then a DIRECT or GDIRECT card is required before the first energy calculation. However, further DIRECT or GDIRECT directives can be given in order to modify specific options or thresholds for particular programs.

The integral-direct implementation in MOLPRO involves three different procedures: (i) Fock matrix evaluation (DFOCK), (ii) integral transformation (DTRAF), and (iii) external exchange operators (DKEXT). Specific options and thresholds exist for all three programs, but it is also possible to specify the most important thresholds by general parameters, which are used as defaults for all programs.

Normally, appropriate default values are automatically used by the program, and in most cases no parameters need to be specified on the DIRECT directive. However, in order to guarantee sufficient accuracy, the default thresholds are quite strict, and in calculations for extended systems larger values might be useful to reduce the CPU time.

The format of the DIRECT directive is

DIRECT, key1=value1, key2=value2...

The following table summarizes the possible keys and their meaning. The default values are given in the subsequent table. In various cases there is a hierarchy of default values. For instance, if THREST_D2EXT is not given, one of the following is used: [THR_D2EXT, THREST_DTRAF, THR_DTRAF, THREST, default]. The list in brackets is checked from left to right, and the first one found in the input is used. default is a default value which depends on the energy threshold and the basis set (the threshold is reduced if the overlap matrix contains very small eigenvalues).

General Options (apply to all programs):

THREST

Integral prescreening threshold. The calculation of an integral shell block is skipped if the product of the largest estimated integral value (based on the Cauchy-Schwarz inequality) and the largest density matrix element contributing to the shell block is smaller than this value. In DTRAF and DKEXT effective density matrices are constructed from the MO coefficients and amplitudes, respectively.

THRINT

Integral prescreening threshold. This applies to the product of the exact (i.e. computed) integral value and a density matrix. This threshold is only used in DTRAF and DKEXT. A shell block of integrals is skipped if the product of the largest integral and the largest element of the effective density matrix contributing to the shell block is smaller than this threshold. If it set negative, no computed integrals will be neglected.

THRPROD

Prescreening threshold for products of integrals and MO-coefficients (DTRAF) or amplitudes (DKEXT). Shell blocks of MO coefficients or amplitudes are neglected if the product of the largest integral in the shell block and the largest coefficient is smaller than this value. If this is set negative, no product screening is performed.

THRMAX

Initial value of the prescreening threshold THREST for DFOCK and DKEXT in iterative methods (SCF, CI, CCSD). If nonzero, it will also be used for DKEXT in MP3 and MP4(SDQ) calculations. The threshold will be reduced to THREST once a certain accuracy has been reached (see VARRED), or latest after MAXRED iterations. In CI and CCSD calculations, also the initial thresholds THRINT_DKEXT and THRPROD_DKEXT are influenced by this value. For a description, see THRMAX_DKEXT. If THRMAX=0, the final thresholds will be used from the beginning in all methods.

SCREEN

Enables or disables prescreening.
SCREEN$\ge 0$: full screening enabled.
SCREEN$\lt 0$: THRPROD is unused. No density screening in direct SCF.
SCREEN$\lt -1$: THRINT is unused.
SCREEN$\lt -2$: THREST is unused.

MAXRED

Maximum number of iterations after which thresholds are reduced to their final values in CI and CCSD calculations. If MAXRED=0, the final thresholds will be used in CI and CCSD from the beginning (same as THRMAX=0, but MAXRED has no effect on DSCF. In the latter case a fixed value of 10 is used.

VARRED

Thresholds are reduced to their final values if the sum of squared amplitude changes is smaller than this value.

SWAP

Enables or disables label swapping in SEWARD. Test purpose only.

Specific options for direct SCF (DFOCK):

THREST_DSCF

Final prescreening threshold in direct SCF. If given, it replaces the value of THREST.

THRMAX_DSCF

Initial prescreening threshold in direct SCF. This is used for the first 7-10 iterations. Once a certain accuracy is reached, the threshold is reduced to THREST_DSCF

SWAP_DFOCK

Enables or disables label swapping in fock matrix calculation (test purpose only).

General options for direct integral transformation (DTRAF):

PAGE_DTRAF

Selects the transformation method.
PAGE_DTRAF=0: use minimum memory algorithm, requiring four integral evaluations.
PAGE_DTRAF=1: use paging algorithm,leading to the minimum CPU time (one integral evaluation for DMP2/LMP2 and two otherwise).

SCREEN_DTRAF

If given, replaces value of ${\tt SCREEN}$ for DTRAF.

MAXSHLQ1_DTRAF

Maximum size of merged shells in the first quarter transformation step (0: not used).

MINSHLQ1_DTRAF

Shells are only merged if their size is smaller than this value (0: not used).

MAXSHLQ2_DTRAF

Maximum size of merged shells in the second quarter transformation step (0: not used).

MINSHLQ2_DTRAF

Shells are only merged if their size is smaller than this value (0: not used).

MAXCEN_DTRAF

Maximum number of centres in merged shells (0: no limit).

PRINT_DTRAF

Print parameter for DTRAF.

General thresholds for all direct integral transformations:

THR_DTRAF

General threshold for DTRAF. If given, this is taken as default value for all thresholds described below.

THREST_DTRAF

AO prescreening threshold for DTRAF.
Defaults: [THR_DTRAF, THREST, default].

THRINT_DTRAF

Integral threshold for DTRAF.
Defaults: [THR_DTRAF, THRINT, default].

THRPROD_DTRAF

Product threshold for DTRAF.
Defaults: [THR_DTRAF, THRPROD, default].

Thresholds specific to direct integral transformations:

THR_D2EXT

General threshold for generation of 2-external integrals. If given, this is used as a default for all D2EXT thresholds described below.

THREST_D2EXT

Prescreening threshold for generation of 2-external integrals.
Defaults: [THR_D2EXT, THREST_DTRAF, THR_DTRAF, THREST, default].

THRINT_D2EXT

Integral threshold for generation of 2-external integrals.
Defaults: [THR_D2EXT, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRPROD_D2EXT

Product threshold for generation of 2-external integrals.
Defaults: [THR_D2EXT, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

THR_D3EXT

General threshold for generation of 3-external integrals. If given, this is used as a default for all D3EXT thresholds described below.

THREST_D3EXT

Prescreening threshold for generation of 3-external integrals.
Defaults: [THR_D3EXT, THREST_DTRAF, THR_DTRAF, THREST, default].

THRINT_D3EXT

Integral threshold for generation of 3-external integrals.
Defaults: [THR_D3EXT, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRPROD_D3EXT

Product threshold for generation of 3-external integrals.
Defaults: [THR_D3EXT, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

THR_D4EXT

General threshold for generation of 4-external integrals. If given, this is used as a default for all D4EXT thresholds described below.

THREST_D4EXT

Prescreening threshold for generation of 4-external integrals.
Defaults: [THR_D4EXT, THREST_DTRAF, THR_DTRAF, THREST, default].

THRINT_D4EXT

Integral threshold for generation of 4-external integrals.
Defaults: [THR_D4EXT, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRPROD_D4EXT

Product threshold for generation of 4-external integrals.
Defaults: [THR_D4EXT, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

THR_DCCSD

General threshold for generalized transformation needed in each CCSD iteration. If given, this is used as a default for THREST_DCCSD, THRINT_DCCSD, and THRPROD_DCCSD described below.

THREST_DCCSD

Prescreening threshold for DCCSD transformation.
Defaults: [THR_DCCSD, THREST_DTRAF, THR_DTRAF, THREST, default].

THRINT_DCCSD

Integral threshold for DCCSD transformation.
Defaults: [THR_DCCSD, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRPROD_DCCSD

Product threshold for DCCSD transformation.
Defaults: [THR_DCCSD, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

THRMAX_DCCSD

Initial value for THREST_DCCSD in CCSD calculations. The threshold will be reduced to THREST_DCCSD once a certain accuracy has been reached (see VARRED), or latest after MAXRED iterations. The initial thresholds THRINT_DCCSD and THRPROD_DCCSD are obtained by multiplying their input (or default) values by THRMAX_DCCSD/THREST_DCCSD, with the restriction that the initial values cannot be smaller than the final ones.

Specific options for direct MP2 (DMP2):

DMP2

Selects the transformation method for direct MP2:
DMP2=$-1$: automatic selection, depending on the available memory.
DMP2=0: use fully direct method for DMP2 (min. two integral evaluations, possibly multipassing, no disk space).
DMP2=1: use semi-direct method for DMP2 (one to four integral evaluations, depending on PAGE_DTRAF).
DMP2=2: use DKEXT to compute exchange operators in DMP2 (one integral evaluation). This is only useful in local DMP2 calculations with many distant pairs.

THR_DMP2

General threshold for generation of 2-external integrals in DMP2. If given, this is used as a default for all DMP2 thresholds described below.

THREST_DMP2

Prescreening threshold for generation of 2-external integrals.
Defaults: [THR_DMP2, THREST_DTRAF, THR_DTRAF, THREST, default].

THRINT_DMP2

Integral threshold for generation of 2-external integrals.
Defaults: [THR_DMP2, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRPROD_DMP2

Product threshold for generation of 2-external integrals
Defaults: [THR_DMP2, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

Specific options for direct local MP2 (LMP2):

DTRAF

Selects the transformation method for direct LMP2:
DTRAF $\geq 0$: generates the 2-external integrals (exchange operators) first in AO basis and transforms these thereafter in a second step to the projected, local basis. The disk storage requirements hence scale cubically with molecular size.
DTRAF $= -1$: generates the 2-external integrals (exchange operators) directly in projected basis. The disk storage requirements hence scale linearly with molecular size. This (together with PAGE_DTRAF = 0) is the recommended algorithm for very large molecules (cf. linear scaling LMP2, chapter 28).
DTRAF $= -2$: alternative algorithm to generate the exchange operators directly in projected basis. Usually, this algorithm turns out to be computationally more expensive than the one selected with DTRAF $= -1$. Note, that neither DTRAF $= -1$ nor DTRAF $= -2$ work in the context of LMP2 gradients.

THR_LMP2

General threshold for generation of 2-external integrals in linear scaling LMP2. If given, this is used as a default for all LMP2 thresholds described below.

THREST_LMP2

Prescreening threshold for generation of 2-external integrals.
Defaults: [THR_LMP2, THREST_DTRAF, THR_DTRAF, THREST, default].

THRQ1_LMP2

Threshold used in the first quarter transformation.
Defaults: [THR_LMP2, THRPROD_DTRAF, THR_DTRAF, THRPROD, default].

THRQ2_LMP2

Threshold used in the second and subsequent quarter transformations.
Defaults: [THR_LMP2, THRINT_DTRAF, THR_DTRAF, THRINT, default].

THRAO_ATTEN

Special threshold for prescreening of attenuated integrals $(\mu \mu \vert \nu \nu)$
Default: THREST_LMP2

Options for integral-direct computation of external exchange operators (DKEXT):

DKEXT

Selects driver for DKEXT.
DKEXT=$-1$: use paging algorithm (minimum memory). This is automatically used if in-core algorithm would need more than one integral pass.
DKEXT=0: use in-core algorithm, no integral triples.
DKEXT=1: use in-core algorithm and integral triples.
DKEXT=2: use in-core algorithm and integral triples if at least two integrals of a triple differ.
DKEXT=3: use in-core algorithm and integral triples if all integrals of a triple differ.

SCREEN_DKEXT

if given, replaces value of ${\tt SCREEN}$ for DKEXT.

MAXSIZE_DKEXT

Largest size of merged shells in DKEXT (0: not used).

MINSIZE_DKEXT

Shells are only merged if their size is smaller than this value. (0: not used).

MAXCEN_DKEXT

Maximum number of centres in merged shells (0: no limit).

SCREEN_DKEXT

Enables of disables screening in DKEXT.

PRINT_DKEXT

Print parameter for DKEXT.

SWAP_DKEXT

Enables of disables label swapping in DKEXT (test purpose only)

MXMBLK_DKEXT

Largest matrix block size in DKEXT (only used with DKEXT$\ge 1$).

Thresholds for integral-direct computation of external exchange operators (DKEXT):

THR_DKEXT

General threshold for DKEXT. If given, this is used as a default for all DKEXT thresholds described below.

THREST_DKEXT

Prescreening threshold for DKEXT.
Defaults: [THR_DKEXT, THREST, default].

THRINT_DKEXT

Integral threshold for DKEXT.
Defaults: [THR_DKEXT, THRINT, default].

THRPROD_DKEXT

Product threshold for DKEXT.
Defaults: [THR_DKEXT, THRPROD, default].

THRMAX_DKEXT

Initial value for THREST_DKEXT in CI, and CCSD calculations. If nonzero. it will also be used for DKEXT in MP3 and MP4(SDQ) calculations. The threshold will be reduced to THREST_DKEXT once a certain accuracy has been reached (see VARRED), or latest after MAXRED iterations. The initial thresholds THRINT_DKEXT and THRPROD_DKEXT are obtained by multiplying their input (or default) values by THRMAX_DKEXT/THREST_DKEXT, with the restriction that the initial values cannot be smaller than the final ones.

For historical reasons, many options have alias names. The following tables summarize the default values for all options and thresholds and also gives possible alias names.

Table 6: Default values and alias names for direct options.

Parameter	Alias	Default value
SCREEN		$1$
MAXRED		$7$
VARRED		1.d-7
SWAP		$1$
SWAP_DFOCK		SWAP
DMP2	DTRAF	$-1$
PAGE_DTRAF	PAGE	$1$
SCREEN_DTRAF		SCREEN
MAXSHLQ1_DTRAF	NSHLQ1	$32$
MINSHLQ1_DTRAF		$0$
MAXSHLQ2_DTRAF	NSHLQ2	$16$
MINSHLQ2_DTRAF		0
MAXCEN_DTRAF		0
PRINT_DTRAF		$-1$
SWAP_DTRAF		SWAP
DKEXT	DRVKEXT	$3$
SCREEN_DKEXT		SCREEN
MAXSIZE_DKEXT		$0$
MINSIZE_DKEXT		$5$
MAXCEN_DKEXT		$1$
PRINT_DKEXT		$-1$
SWAP_DKEXT		SWAP
MXMBLK_DKEXT		depends on hardware (-B parameter on molpro command)

Table 7: Default thresholds and alias names for direct calculations

Parameter	Alias	Default value
THREST	THRAO	$\min(\Delta E \cdot 1.d-2,1.d-9)^{a,b}$
THRINT	THRSO	$\min(\Delta E \cdot 1.d-2,1.d-9)^{a,b}$
THRPROD	THRP	$\min(\Delta E \cdot 1.d-3,1.d-10)^{a,b}$
THRMAX		1.d-8$^b$
THREST_DSCF	THRDSCF	$\le$ 1.d-10 (depending on accuracy and basis set)
THRMAX_DSCF	THRDSCF_MAX	THRMAX
THR_DTRAF	THRDTRAF
THREST_DTRAF	THRAO_DTRAF	[THR_DTRAF, THREST]
THRINT_DTRAF	THRAO_DTRAF	[THR_DTRAF, THRINT]
THRPROD_DTRAF	THRP_DTRAF	[THR_DTRAF, THRPROD]
THR_D2EXT	THR2EXT	THR_DTRAF
THREST_D2EXT	THRAO_D2EXT	[THR_D2EXT, THREST_DTRAF]
THRINT_D2EXT	THRSO_D2EXT	[THR_D2EXT, THRINT_DTRAF]
THRPROD_D2EXT	THRP_D2EXT	[THR_D2EXT, THRPROD_DTRAF]
THR_D3EXT	THR3EXT	THR_DTRAF
THREST_D3EXT	THRAO_D3EXT	[THR_D3EXT, THREST_DTRAF]
THRINT_D3EXT	THRSO_D3EXT	[THR_D3EXT, THRINT_DTRAF]
THRPROD_D3EXT	THRP_D3EXT	[THR_D3EXT, THRPROD_DTRAF]
THR_D4EXT	THR4EXT	THR_DTRAF
THREST_D4EXT	THRAO_D4EXT	[THR_D4EXT, THREST_DTRAF]
THRINT_D4EXT	THRSO_D4EXT	[THR_D4EXT, THRINT_DTRAF]
THRPROD_D4EXT	THRP_D4EXT	[THR_D4EXT, THRPROD_DTRAF]
THR_DCCSD	THRCCSD	THR_DTRAF
THREST_DCCSD	THRAO_DCCSD	[THR_DCCSD, THREST_DTRAF]
THRINT_DCCSD	THRSO_DCCSD	[THR_DCCSD, THRINT_DTRAF]
THRPROD_DCCSD	THRP_DCCSD	[THR_DCCSD, THRPROD_DTRAF]
THRMAX_DCCSD	THRMAX_DTRAF	THRMAX
THR_DMP2	THRDMP2	THR_DTRAF
THREST_DMP2	THRAO_DMP2	[THR_DMP2, THREST_DTRAF, default$^c$]
THRINT_DMP2	THRSO_DMP2	[THR_DMP2, THRINT_DTRAF, default$^c$]
THRPROD_DMP2	THRP_DMP2	[THR_DMP2, THRPROD_DTRAF, default$^c$]
THR_LMP2	THRLMP2	THR_DTRAF
THREST_LMP2	THRAO_LMP2	[THR_LMP2, THREST_DTRAF, default$^c$]
THRQ1_LMP2	THRQ1	[THR_LMP2, THRPROD_DTRAF, default$^c$]
THRQ2_LMP2	THRQ2	[THR_LMP2, THRINT_DTRAF, default$^c$]
THRAO_ATTEN ]	THRATTEN	THREST_LMP2
THR_DKEXT	THRKEXT
THREST_DKEXT	THRAO_DKEXT	[THR_DKEXT, THREST]
THRINT_DKEXT	THRSO_DKEXT	[THR_DKEXT, THRINT]
THRPROD_DKEXT	THRP_DKEXT	[THR_DKEXT, THRPROD]
THRMAX_DKEXT		THRMAX

a) $\Delta E$ is the requested accuracy in the energy (default 1.d-6).
b) The thresholds are reduced if the overlap matrix has small eigenvalues.
c) The default thresholds for DMP2 and LMP2 are $0.1 \cdot {\Delta E}$.