# MP2

### From NWChem

# MP2

There are (at least) three algorithms within NWChem that compute the Møller-Plesset (or many-body) perturbation theory second-order correction^{[1]} to the Hartree-Fock energy (MP2). They vary in capability, the size of system that can be treated and use of other approximations

- Semi-direct -- this is recommended for most large applications (up to about 2800 basis functions), especially on the IBM SP and other machines with significant disk I/O capability. Partially transformed integrals are stored on disk, multi-passing as necessary. RHF and UHF references may be treated including computation of analytic derivatives. This is selected by specifying mp2 on the task directive, e.g.

TASK MP2

- Fully-direct
^{[2]}-- this is of utility if only limited I/O resources are available (up to about 2800 functions). Only RHF references and energies are available. This is selected by specifying direct_mp2 on the task directive, e.g.

TASK DIRECT_MP2

- Resolution of the identity (RI) approximation MP2 (RI-MP2)
^{[3]}-- this uses the RI approximation and is therefore only exact in the limit of a complete fitting basis. However, with some care, high accuracy may be obtained with relatively modest fitting basis sets. An RI-MP2 calculation can cost over 40 times less than the corresponding exact MP2 calculation. RHF and UHF references with only energies are available. This is selected by specifying rimp2 on the task directive, e.g.,

TASK RIMP2

All three MP2 tasks share the same input block.

MP2 [FREEZE [[core] (atomic || <integer nfzc default 0>)] \ [virtual <integer nfzv default 0>]] [TIGHT] [PRINT] [NOPRINT] [VECTORS <string filename default scf-output-vectors> \ [swap [(alpha||beta)] <integer pair-list>] ] [RIAPPROX <string riapprox default V>] [FILE3C <string filename default $file_prefix$.mo3cint">] [SCRATCHDISK <integer>] END

## FREEZE -- Freezing orbitals

All MP2 modules support frozen core orbitals, however, only the direct MP2 and RI-MP2 modules support frozen virtual orbitals.

By default, no orbitals are frozen. The atomic keyword causes orbitals to be frozen according to the rules in the table below. Note that no orbitals are frozen on atoms on which the nuclear charge has been modified either by the user or due to the presence of an ECP. The actual input would be

freeze atomic

For example, in a calculation on *S**i*(*O**H*)_{2}, by default the lowest seven orbitals would be frozen (the oxygen 1*s*, and the silicon 1*s*, 2*s* and 2*p*).

Period | Elements | Core Orbitals | Number of Core |

0 | H - He | - | 0 |

1 | Li - Ne | 1s
| 1 |

2 | Na - Ar | 1s2s2p
| 5 |

3 | K - Kr | 1s2s2p3s3p
| 9 |

4 | Rb - Xe | 1s2s2p3s3p4s3d4p
| 18 |

5 | Cs - Rn | 1s2s2p3s3p4s3d4p5s4d5p
| 27 |

6 | Fr - Lr | 1s2s2p3s3p4s3d4p5s4d5p6s4f5d6p
| 43 |

Caution: The rule for freezing orbitals "by atoms" are rather unsophisticated: the number of orbitals to be frozen is computed from the Table 16.1 by summing the number of core orbitals in each atom present. The corresponding number of lowest-energy orbitals are frozen -- if for some reason the actual core orbitals are not the lowest lying, then correct results will not be obtained. From limited experience, it seems that special attention should be paid to systems including third- and higher- period atoms.

The user may also specify the number of orbitals to be frozen by atom. Following the *S**i*(*O**H*)_{2} example, the user could specify

freeze atomic O 1 Si 3

In this case only the lowest four orbitals would be frozen. If the user does not specify the orbitals by atom, the rules default to Table 16.1.

Caution: The system does not check for a valid number of orbitals per atom. If the user specifies to freeze more orbitals then are available for the atom, the system will not catch the error. The user must specify a logical number of orbitals to be frozen for the atom.

The FREEZE directive may also be used to specify the number of core orbitals to freeze. For instance, to freeze the first 10 orbitals

freeze 10

or equivalently, using the optional keyword core

freeze core 10

Again, note that if the 10 orbitals to be frozen do not correspond to the first 10 orbitals, then the swap keyword of the VECTORS directive must be used to order the input orbitals correctly (MO vectors).

To freeze the highest virtual orbitals, use the virtual keyword. For instance, to freeze the top 5 virtuals

freeze virtual 5

Again, note that this only works for the direct-MP2 and RI-MP2 energy codes.

## TIGHT -- Increased precision

The TIGHT directive can be used to increase the precision in the MP2 energy and gradients.

By default the MP2 gradient package should compute energies accurate to better than a micro-Hartree, and gradients accurate to about five decimal places (atomic units). However, if there is significant linear dependence in the basis set the precision might not be this good. Also, for computing very accurate geometries or numerical frequencies, greater precision may be desirable.

This option increases the precision to which both the SCF (from 10^{ − 6} to 10^{ − 8}) and CPHF (from 10^{ − 4} to 10^{ − 6}) are solved, and also tightens thresholds for computation of the AO and MO integrals (from 10^{ − 9} to 10^{ − 11}) within the MP2 code.

## SCRATCHDISK -- Limiting I/O usage

This directive - used only in the semi-direct algorithm - allows to limit the per process disk usage. Mandatory argument for this keyword is the maximum number of MBytes. For example, the following input line

scratchdisk 512

puts an upper limit of 512 MBytes to the semi-direct MP2 usage of disk (again, on a per process base).

## PRINT and NOPRINT

The standard print control options are recognized. The list of recognized names are given in the table below.

Item | Print Level | Description |

RI-MP2
| ||

"2/3 ints" | debug | Partial 3-center integrals |

"3c ints" | debug | MO 3-center integrals |

"4c ints b" | debug | "B" matrix with approx. 4c integrals |

"4c ints" | debug | Approximate 4-center integrals |

"amplitudes" | debug | "B" matrix with denominators |

"basis" | high | |

"fit xf" | debug | Transformation for fitting basis |

"geombas" | debug | Detailed basis map info |

"geometry" | high | |

"information" | low | General information about calc. |

"integral i/o" | high | File size information |

"mo ints" | debug | |

"pair energies" | debug | (working only in direct_mp2) |

"partial pair energies" | debug | Pair energy matrix each time it is updated |

"progress reports" | default | Report completion of time-consuming steps |

"reference" | high | Details about reference wavefunction |

"warnings" | low | Non-fatal warnings |

## VECTORS -- MO vectors

All of the (supported) MP2 modules require use of converged canonical SCF (RHF or UHF) orbitals for correct results. The vectors are by default obtained from the preceding SCF calculation, but it is possible to specify a different source using the VECTORS directive. For instance, to obtain vectors from the file /tmp/h2o.movecs, use the directive

vectors /tmp/h2o.movecs

As noted above (FREEZE) if the SCF orbitals are not in the correct order, it is necessary to permute the input orbitals using the swap keyword of the VECTORS directive. For instance, if it is desired to freeze a total six orbitals corresponding to the SCF orbitals 1-5, and 7, it is necessary to swap orbital 7 into the 6th position. This is accomplished by

vectors swap 6 7

The swap capability is examined in more detail in Input/output of MO vectors.

## RI-MP2 fitting basis

The RI-MP2 method requires a fitting basis, which must be specified with the name "ri-mp2 basis" (see Basis). For instance,

basis "ri-mp2 basis" O s; 10000.0 1 O s; 1000.0 1 O s; 100.0 1 ... end

Alternatively, using a standard capability of basis sets (Basis) another named basis may be associated with the fitting basis. For instance, the following input specifies a basis with the name "small fitting basis" and then defines this to be the "ri-mp2 basis".

basis "small fitting basis" H s; 10 1 H s; 3 1 H s; 1 1 H s; 0.1 1 H s; 0.01 1 end

set "ri-mp2 basis" "small fitting basis"

## FILE3C -- RI-MP2 3-center integral filename

The default name for the file used to store the transformed 3-center integrals is "$file_prefix$.mo3cint" in the scratch directory. This may be overridden using the FILE3C directive. For instance, to specify the file /scratch/h2o.3c, use this directive

file3c /scratch/h2o.3c

## RIAPPROX -- RI-MP2 Approximation

The type of RI approximation used in the RI-MP2 calculation is controlled by means of the RIAPPROX directive. The two possible values are V and SVS (case sensitive), which correspond to the approximations with the same names described by Vahtras et al.^{[4]}. The default is V.

## Advanced options for RI-MP2

These options, which functioned at the time of writing, are not currently supported.

### Control of linear dependence

Construction of the RI fit requires the inversion of a matrix of fitting basis integrals which is carried out via diagonalization. If the fitting basis includes near linear dependencies, there will be small eigenvalues which can ultimately lead to non-physical RI-MP2 correlation energies. Eigenvectors of the fitting matrix are discarded if the corresponding eigenvalue is less than *m**i**n**e**v**a**l* which defaults to 10^{ − 8}. This parameter may be changed by setting the a parameter in the database. For instance, to set it to 10^{ − 10}

set "mp2:fit min eval" 1e-10

### Reference Spin Mapping for RI-MP2 Calculations

The user has the option of specifying that the RI-MP2 calculations are to be done with variations of the SCF reference wavefunction. This is accomplished with a SET directive of the form,

set "mp2:reference spin mapping" <integer array default 0>

Each element specified for array is the SCF spin case to be used for the corresponding spin case of the correlated calculation. The number of elements set determines the overall type of correlated calculation to be performed. The default is to use the unadulterated SCF reference wavefunction.

For example, to perform a spin-unrestricted calculation (two elements) using the alpha spin orbitals (spin case 1) from the reference for both of the correlated reference spin cases, the SET directive would be as follows,

set "mp2:reference spin mapping" 1 1

The SCF calculation to produce the reference wavefunction could be either RHF or UHF in this case.

The SET directive for a similar case, but this time using the beta-spin SCF orbitals for both correlated spin cases, is as follows,

set "mp2:reference spin mapping" 2 2

The SCF reference calculation must be UHF in this case.

The SET directive for a spin-restricted calculation (one element) from the beta-spin SCF orbitals using this option is as follows,

set "mp2:reference spin mapping" 2

The SET directive for a spin-unrestricted calculation with the spins flipped from the original SCF reference wavefunction is as follows,

set "mp2:reference spin mapping" 2 1

### Batch Sizes for the RI-MP2 Calculation

The user can control the size of each batch in the transformation and energy evaluation in the MP2 calculation, and consequently the memory requirements and number of passes required. This is done using two SET directives of the following form,

set "mp2:transformation batch size" <integer size default -1> set "mp2:energy batch size" <integer isize jsize default -1 -1>

The default is for the code to determine the batch size based on the available memory. Should there be problems with the program-determined batch sizes, these variables allow the user to override them. The program will always use the smaller of the user's value of these entries and the internally computed batch size.

The transformation batch size computed in the code is the number of occupied orbitals in the *(occ vir|fit)* three-center integrals to be produced at a time. If this entry is less than the number of occupied orbitals in the system, the transformation will require multiple passes through the two-electron integrals. The memory requirements of this stage are two global arrays of dimension *<batch size> x vir x fit* with the "fit" dimension distributed across all processors (on shell-block boundaries). The compromise here is memory space versus multiple integral evaluations.

The energy evaluation batch sizes are computed in the code from the number of occupied orbitals in the two sets of three-center integrals to be multiplied together to produce a matrix of approximate four-center integrals. Two blocks of integrals of dimension *(<batch isize> x vir)* and *(<batch jsize> x vir)* by fit are read in from disk and multiplied together to produce *<batch isize> <batch jsize> vir^2* approximate integrals. The compromise here is performance of the distributed matrix multiplication (which requires large matrices) versus memory space.

### Energy Memory Allocation Mode: RI-MP2 Calculation

The user must choose a strategy for the memory allocation in the energy evaluation phase of the RI-MP2 calculation, either by minimizing the amount of I/O, or minimizing the amount of computation. This can be accomplished using a SET directive of the form,

set "mp2:energy mem minimize" <string mem_opt default I>

A value of I entered for the string mem_opt means that a strategy to minimize I/O will be employed. A value of C tells the code to use a strategy that minimizes computation.

When the option to minimize I/O is selected, the block sizes are made as large as possible so that the total number of passes through the integral files is as small as possible. When the option to minimize computation is selected, the blocks are chosen as close to square as possible so that permutational symmetry in the energy evaluation can be used most effectively.

### Local Memory Usage in Three-Center Transformation

For most applications, the code will be able to size the blocks without help from the user. Therefore, it is unlikely that users will have any reason to specify values for these entries except when doing very particular performance measurements.

The size of xf3ci:AO 1 batch size is the most important of the three, in terms of the effect on performance.

Local memory usage in the first two steps of the transformation is controlled in the RI-MP2 calculation using the following SET directives,

set "xf3ci:AO 1 batch size" <integer max> set "xf3ci:AO 2 batch size" <integer max> set "xf3ci:fit batch size" <integer max>

The size of the local arrays determines the sizes of the two matrix multiplications. These entries set limits on the size of blocks to be used in each index. The listing above is in order of importance of the parameters to performance, with xf3ci:AO 1 batch size being most important.

Note that these entries are only upper bounds and that the program will size the blocks according to what it determines as the best usage of the available local memory. The absolute maximum for a block size is the number of functions in the AO basis, or the number of fitting basis functions on a node. The absolute minimum value for block size is the size of the largest shell in the appropriate basis. Batch size entries specified for max that are larger than these limits are automatically reset to an appropriate value.

## One-electron properties and natural orbitals

If an MP2 energy gradient is computed, all contributions are available to form the MP2 linear-response density. This is the density that when contracted with any spin-free, one-electron operator yields the associated property defined as the derivative of the energy. Thus, the reported MP2 dipole moment is the derivative of the energy w.r.t. an external magnetic field and is not the expectation value of the operator over the wavefunction. Only dipole moments are printed by the MP2 gradient code, but natural orbitals are produced and stored in the permanent directory with a file extension of ".mp2nos". These may be fed into the property package to compute more general properties. Note that the MP2 linear response density matrix is not necessarily positive definite so it is not unusual to see a few small negative natural orbital occupation numbers.

## References

- ↑ Møller, C. and Plesset, M.S. (1934) "Note on an approximation treatment for many-electron systems",
*Physical Review***46**618-622, doi:10.1103/PhysRev.46.618. - ↑ Wong, A.T.; Harrison, R.J. and Rendell, A.P. (1996) "Parallel direct four-index transformations",
*Theoretica Chimica Acta***93**317-331, doi:10.1007/BF01129213 - ↑ Bernholdt, D.E. and Harrison, R.J. (1996) "Large-scale correlated electronic structure calculations: the RI-MP2 method on parallel computers",
*Chemical Physics Letters***250**(5-6) 477-484, doi:10.1016/0009-2614(96)00054-1 - ↑ Vahtras, O.; Almlöf, J. and Feyereisen, M. W. (1993) "Integral approximations for LCAO-SCF calculations",
*Chem. Phys. Lett.***213**, 514-518, doi: 10.1016/0009-2614(93)89151-7