Multi-reference perturbation theory (MRPT)

Modules: mrpt

Introduction

A second-order perturbative energy correction can be added on top of a multireference wave function. The MRPT module in PySCF supports the second-order N-electron valence state perturbation theory (NEVPT2) [1] using the strongly contracted (SC) internal contraction scheme, [2][3] which is an intruder-state-free MRPT. SC-NEVPT2 can be applied to CASCI/CASSCF wave functions produced by the FCI or DMRG solvers. [4] The number of the CI root needs to be specified for state-specific NEVPT2 calculations with mrpt.NEVPT(mc,root=Root_ID). By default, the NEVPT2 calculation is performed for the lowest root, Root_ID=0.

A simple example of SC-NEVPT2 calculations with the FCI and DMRG solvers is given in examples/mrpt/03-dmrg_nevpt2.py

#!/usr/bin/env python
#
# Author: Sheng Guo <shengg@princeton.edu>
#         Qiming Sun <osirpt.sun@gmail.com>
#         Seunghoon Lee <seunghoonlee89@gmail.com>
#

'''
DMRG-CASCI then DMRG-NEVPT2 calculation.

There are two NEVPT2 implementations available for DMRG Block program.  The slow
version (default) strictly follows the formula presented in JCP, 117(2002), 9138
in which the 4-particle density matrix is explictly computed.  Typically 26
orbitals is the upper limit of the slow version due to the large requirements
on the memory usage.  The fast version employs the so called MPS-pertuber
technique.  It is able to handle much larger systems, up to about 30 orbitals.
'''

from pyscf import gto, scf, mcscf, mrpt, dmrgscf
mol = gto.M(atom = [['H', (0.,0.,i-3.5)] for i in range(8)],basis = 'sto-3g',symmetry='d2h')
m = scf.RHF(mol).run()

##############################################################################
#
# FCI-based CASCI + NEVPT2.  Two roots are computed.  mc.ci holds the two CI
# wave functions.  Root ID needs to be specified for the state-specific NEVPT2
# calculation.  By default the lowest root is computed.
#
##############################################################################
mc = mcscf.CASCI(m, 4, 4)
mc.fcisolver.nroots = 2
mc.fix_spin_(shift=.5, ss=0)
mc.kernel()

ci_nevpt_e1 = mrpt.NEVPT(mc, root=0).kernel()
ci_nevpt_e2 = mrpt.NEVPT(mc, root=1).kernel()

print('FCI NEVPT correlation E = %.15g %.15g' % (ci_nevpt_e1, ci_nevpt_e2))

##################################################
#
# DMRG-NEVPT2 fast version
# Use compressed MPS as perturber functions for SC-NEVPT2.
# 4-particle density matrix is not computed.
#
##################################################

mc = mcscf.CASCI(m, 4, 4)
dmrgscf.settings.MPIPREFIX =''
mc.fcisolver = dmrgscf.DMRGCI(mol, maxM=200)
mc.fcisolver.nroots = 2
mc.kernel()

mps_nevpt_e1 = mrpt.NEVPT(mc, root=0).compress_approx(maxM=100).kernel()
mps_nevpt_e2 = mrpt.NEVPT(mc, root=1).compress_approx(maxM=100).kernel()

print('MPS NEVPT correlation E = %.15g %.15g' % (mps_nevpt_e1, mps_nevpt_e2,))

which outputs

FCI NEVPT correlation E = -0.0655576579894365 -0.0913916717329482
MPS NEVPT correlation E = -0.0655570680554582 -0.0913913618723217

namely, the second-order correlation energies for the ground and the first-excited states with the FCI solver, and those with the DMRG solver.

Compressed Perturber Functions

The bottleneck in SC-NEVPT2 is the evaluation of the energies of the perturber functions, where up to the 4-particle reduced density matrix (4-RDM) appears. In DMRG-SC-NEVPT2, this evaluation is done with the compressed bond dimension (M’) which is smaller than the bond dimension in the DMRG energy optimization to avoid the bottleneck. It can be specified by mrpt.NEVPT(mc,root=Root_ID).compress_approx(maxM=M').

More information can be found in Reference [4]

References

1

Celestino Angeli, Renzo Cimiraglia, S Evangelisti, T Leininger, and J-P Malrieu. Introduction of n-electron valence states for multireference perturbation theory. J. Chem. Phys., 114(23):10252–10264, 2001. doi:10.1063/1.1361246.

2

Celestino Angeli, Renzo Cimiraglia, and Jean-Paul Malrieu. N-electron valence state perturbation theory: a fast implementation of the strongly contracted variant. Chem. Phys. Lett., 350(3-4):297–305, 2001. doi:10.1016/S0009-2614(01)01303-3.

3

Celestino Angeli, Renzo Cimiraglia, and Jean-Paul Malrieu. N-electron valence state perturbation theory: a spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants. J. Chem. Phys., 117(20):9138–9153, 2002. doi:10.1063/1.1515317.

4(1,2)

Sheng Guo, Mark A Watson, Weifeng Hu, Qiming Sun, and Garnet Kin-Lic Chan. N-electron valence state perturbation theory based on a density matrix renormalization group reference function, with applications to the chromium dimer and a trimer model of poly (p-phenylenevinylene). J. Chem. Theory Comput., 12(4):1583–1591, 2016. doi:10.1021/acs.jctc.5b01225.