Davidson size-consistency corrections for truncated CI wave functions¶

Background¶

A serious deficiency of truncated CI expansions, including the most popular ones, namely (multi-reference; MR)CI-SD is the lack of size-consistency; i.e., the energy of the system does not scale properly with the system size. Besides corrections to the CI equations which will improve the energy and the wave function, a posteriori corrections (most often referred to as Davidson-type corrections or +Q) to the energy have become quite popular. A comprehensive overview on this topic has been given recently in a review paper on multiconfigurational and multirefernce methods. Starting with Dirac14 we have implemented a number of different flavors of the latter a posteriori corrections to the energy for the modules KRCI and LUCITA. The nomenclature and detailed equations for each correction is explained in great detail in the above review article (see chapter 2.1.3.1).

Example: Water¶

Following the example for water in the review article we need to run two calculations. Although it’s possible to do it in one step we will split the task in two.

Step 1: Get reference wave function and configuration(s)¶

We first start with a multiconfigurational SCF calculation for water with a CAS(4,4) space using a cc-pVDZ basis set. This will yield a suitable reference wave function for the subsequent CI. The MCSCF module will in addition automatically create a file named refvec.luci which contains the reference configurations needed later for the +Q corrections. Our CASSCF input scf-casscf.inp is

**DIRAC
.TITLE
qcorr
.WAVE FUNCTION
**INTEGRALS
.UNCONTRACTED
**HAMILTONIAN
.DOSSSS
.X2C
**WAVE FUNCTION
.SCF
.KRMCSCF
*SCF
.CLOSED SHELL
10
*KRMCSCF
.CI PROGRAM
LUCIAREL
.INACTIVE
3
.GAS SHELLS
1
4 4 / 4
.SYMMETRY
1
.THRESH
1.0d-04
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
**END OF


The molecular structure is located in h2o.xyz

3
H2O - geometry from Chem. Rev. paper MCSCF/CI/QCORR (2012).
O 0.00000000 0.00000000 -0.11976444
H 0.00000000 -0.80201121 0.43557446
H 0.00000000 0.80201121 0.43557446


Suppose that we have run the CASSCF calculation using:

pam --inp=scf-casscf.inp  --mol=h2o.xyz --outkrmc --get="refvec.luci"


where we keep the MCSCF coefficients (in KRMCSCF) and the reference vectors stored in refvec.luci we are now in position to proceed to Step 2.

Step 2: Run the MRCISD calculation and compute the +Q corrections¶

In the second step we now run a MRCISD calculation using the reference wave function coefficients from the previous step as starting point and compute the +Q corrections to the MRCISD energy. Using the same molecular input structure we run the MRCISD+Q calculation using the input file krci-q.inp

**DIRAC
.TITLE
qcorr
.WAVE FUNCTION
**INTEGRALS
.UNCONTRACTED
**HAMILTONIAN
.X2C
.DOSSSS
**WAVE FUNCTION
.KR CI
*KRCICALC
.CI PROGRAM
LUCIAREL
.INACTIVE
0
.GAS SHELLS
3
4 6 / 3
8 10 / 4
10 10 / 9
.CIROOTS
1 1
.MAX CI
200
.MXCIVE
36
.NOOCCN
*QCORR
.REFFILE
.CVORB
3 0
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
**END OF


with the following start command:

pam --inp=scf-casscf.inp  --mol=h2o.xyz --inkrmc --put="refvec.luci"


We will find a summary of different +Q corrections printed at the end of the output:

  The acronyms for each Q correction below  refer to:
Edc:  Davidson correction              [Eq. (34)]
Erdc: renormalized Davidson correction [Eq. (36)]
Epc: modified Pople correction        [Eq. (39)]
Emc : Meissner correction              [Eq. (41)]

Q corrections for state   1 with c0^2 according to Eq. (43)
------------------------------------------------------------------------------------------------------------------------
model              Edc             Erdc             Epc              Emc        Emc (Nact - 2e)
CI+Q energy           -76.221619       -76.221689       -76.221187       -76.220742       -76.220525
------------------------------------------------------------------------------------------------------------------------

Q corrections for state   1 with c0^2 according to Eq. (44)
------------------------------------------------------------------------------------------------------------------------
model              Edc             Erdc             Epc              Emc        Emc (Nact - 2e)
CI+Q energy           -76.221645       -76.221716       -76.221209       -76.220758       -76.220539
------------------------------------------------------------------------------------------------------------------------

`

where a detailed account of the general performance of the different +Q corrections (and which other QC programs have implemented which +Q correction by default) is given in chapter 2.1.3.1 of the above given review article. The equations references given in the output match the equation labels given in chapter 2.1.3.1. We strongly recommend the interested user to carefully read this chapter 2.1.3.1.