# Integration

Before starting any energy calculations, the geometry and basis set must be defined in GEOMETRY and BASIS blocks, respectively. By default, two electron integrals are evaluated once and stored on disk. This behaviour may be overridden by using the input command gdirect (see section INTEGRAL-DIRECT CALCULATIONS (GDIRECT)) to force evaluation of integrals on the fly. Molpro checks if the one-and two-electron integrals are available for the current basis set and geometry, automatically computing them if necessary. The program also recognizes automatically if only the nuclear charges have been changed, as is the case in counterpoise calculations. In this case, the two-electron integrals are not recomputed.

By default a point charge nuclear model is used for all atoms. Alternatively, a Gaussian nuclear model can be used by setting

SET,FNUC=1
before the first energy evaluation (a value of 0 corresponds to a point charge nucleus). Alternatively, this also can be given as an option to the INT command:

INT,FNUC=1

If the integrals are stored on disk, immediately after evaluation they are sorted into complete symmetry-packed matrices, so that later program modules that use them can do so as efficiently as possible. As discussed above, it is normally not necessary to call the integral and sorting programs explicitly, but sometimes additional options are desired, and can be specified using the INT command, which should appear after geometry and basis specifications, and before any commands to evaluate an energy.

INT, [[NO]SORT,] [SPRI=value]

SORT, [SPRI=value]

INT,NOSORT;SORT can be used to explicitly separate the integral evaluation and sorting steps, for example to collect separate timing data. With value set to more than 1 in the SPRI option, all the two-electron integrals are printed.

The detailed options for the integral sort can be specified using the AOINT parameter set, using the input form

AOINT, key1=value1, key2=value2, $\dots$

AOINT can be used with or without an explicit INT command.

The following summarizes the possible keys, together with their meaning, and default values.

• c_final Integer specifying the compression algorithm to be used for the final sorted integrals. Possible values are 0 (no compression), 1 (compression using 1, 2, 4 or 8-byte values), 2 (2, 4 or 8 bytes), 4 (4, 8 bytes) and 8. Default: 0
• c_sort1 Integer specifying the compression algorithm for the intermediate file during the sort. Default: 0
• c_seward Integer specifying the format of label tagging and compression written by the integral program and read by the sort program. Default: 0
• compress Overall compression; c_final, c_seward and c_sort1 are forced internally to be not less than this parameter. Default: 1
• thresh Real giving the truncation threshold for compression. Default: $0.0$, which means use the integral evaluation threshold (GTHRESH,TWOINT)
• io String specifying how the sorted integrals are written. Possible values are molpro (standard Molpro record on file 1) and eaf (Exclusive-access file). eaf is permissible only if the program has been configured for MPP usage, and at present molpro is implemented only for serial execution. molpro is required if the integrals are to be used in a restart job. For maximum efficiency on a parallel machine, eaf should be used, since in that case the integrals are distributed on separate processor-local files.

For backward-compatibility purposes, two convenience commands are also defined: COMPRESS is equivalent to AOINT,COMPRESS=1, and NOCOMPRESS is equivalent to AOINT,COMPRESS=0.

It is possible to import the second-quantised hamiltonian completely from outside Molpro. In order to do so, it is necessary to set up a job that simulates the desired calculation by having a basis set of exactly the same dimensions as the one to be imported. One can then import the hamiltonian using the command

HAMILTONIAN,filename

filename is the name of a file that contains the hamiltonian in FCIDUMP format, which can be produced using Molpro’s {FCI,DUMP=} facility, or by another method.

Note that this facility is fragile, and is limited to energy-only calculations. Attempts to calculate gradients or other properties will inevitably fail. At present, the implementation does not support the use of point-group symmetry.

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).

Most methods implemented in MOLPRO 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. Exceptions are currently full CI (FCI), perturbative triple excitations (T), UMP2, RMP2, CPP, MRCI-F12, and RS2-F12. 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 is 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 PAO-based local correlation treatments).
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 $(\mu \mu | \nu \nu)$

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.

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)
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 $0.1 \cdot {\Delta E}$.

examples/h2o_direct.inp
$method=[hf,mp2,ccsd,qci,bccd,multi,mrci,acpf,rs3] !some methods basis=vdz !basis geometry={o;h1,o,r;h2,o,r,h1,theta} !geometry gdirect !direct option r=1 ang,theta=104 !bond length and angle do i=1,#method !loop over methods$method(i)                                                     !run method(i)
e(i)=energy                                                    !save results in variables
dip(i)=dmz
enddo
table,method,e,dip                                             !print table of results

This job produces the following table:

 METHOD       E            DIP
HF       -76.02145798   0.82747348
MP2      -76.22620591   0.00000000
CCSD     -76.23580191   0.00000000
QCI      -76.23596211   0.00000000
BCCD     -76.23565813   0.00000000
MULTI    -76.07843443   0.76283026
MRCI     -76.23369819   0.76875001
ACPF     -76.23820180   0.76872802
RS3      -76.23549448   0.75869972