## Multi-geminal wave functions

General discussion of the Cambridge quantum Monte Carlo code CASINO; how to install and setup; how to use it; what it does; applications.

### Re: Multi-geminal wave functions

Pablo_Lopez_Rios wrote:Patch now available in current beta.

Best,
Pablo

Hello Pablo, thanks for the opportunity to test the calculations with geminals ansatz.

I chose the simplest systems and performed the calculations in the qchem program.

1. Li-atom in cc-pVDZ basis, WFN consists of 1 geminal orbitals and an orbital filled with an unpaired electron, so QCHEM input/output and CASINO input/output in attachment.

Final energy = -7.432605555957891

Energy decomposition, spin= TRUE

Nuclear Attr. Energy = 0.0000000000

One-El. Pot. Energy = -17.1450277592
One-El. Kin. Energy = 7.4317293525
Tot. Intra-gem Energy = -9.2971415985
Inter-gem Coul. Energy = 2.6905114013
Inter-gem Exch. Energy = -0.8259753588
Correlation Energy = -0.0004001996
VAR verifying total -7.432605555957891 error: -0.000000000000001

Geminal occupations, energies and coefficients.

Geminal 1 E = -3.387669043962796
2.00[s],
0.99996729 -0.00464321 -0.00464307 -0.00464305 -0.00048555 -0.00031930 -0.00031926 -0.00031921 -0.00031917 -0.00031659 -0.00001423 -0.00001403 0.00001401.

Occupations and energies of unpaired orbitals.
1.00[s],
-2.18040047

Unfortunately CASINO cannot accept an unpaired electron on second orbital and returns an error: Kinetic energy test failed: analytical derivatives misbehave.

2. Be-atom in cc-pVDZ basis, WFN consists of 2 geminals strongly ortogonal, so QCHEM input/output and CASINO input/output in attachment.

Final energy = -14.617016246569332

Energy decomposition, spin= TRUE

Nuclear Attr. Energy = 0.0000000000

One-El. Pot. Energy = -33.6940528313
One-El. Kin. Energy = 14.6150473252
Tot. Intra-gem Energy = -16.5188194534
Inter-gem Coul. Energy = 1.9430696836
Inter-gem Exch. Energy = -0.0412664768
Correlation Energy = -0.1214694892
VAR verifying total -14.617016246569332 error: 0.000000000000017

Geminal occupations, energies and coefficients.

Geminal 1 E = -11.707720511899904
2.00[s],
0.99999849 -0.00100242 -0.00100221 -0.00100180.

Geminal 2 E = -1.007492527867571
1.82[s],
0.95271649 -0.17380094 -0.17379549 -0.17379340 -0.03715659 -0.00818632 -0.00818536 -0.00818511 -0.00818199 -0.00818198

each virtual orbital belongs either to the first or second geminal, this results in orthogonality, and I manually distributed virtual orbitals based on the QCHEM output file, however, the energy does not coincide with that found in QCHEM output file.
I will be very grateful if you describe the algorithm for constructing a wave function (GEMINAL section) for two orthogonal geminal.

if changes were made to a separate git branch it would be much easier to test, without breaking master branch.

Attachments Be.tgz Li.tgz
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### Re: Multi-geminal wave functions

Vladimir_Konjkov wrote:Li-atom in cc-pVDZ basis, WFN consists of 1 geminal orbitals and an orbital filled with an unpaired electron

Unpaired electrons are indeed not supported; the geminals are supposed to be determinants of square matrices of size Nup x Ndown which must therefore satisfy Nup = Ndown. If you could point at a description of how geminals are defined for systems with Nup /= Ndown I might be able to implement this.

Vladimir_Konjkov wrote:Be-atom in cc-pVDZ basis, WFN consists of 2 geminals strongly ortogonal,

You could start by trying each of these geminals separately to see if individual energies match those from QCHEM. In the CASINO input files you have set the coefficients of both geminals to one, but I doubt this is optimal. Does QCHEM provide the coefficients? If not, it should be easy to fix c_1=1 and optimize c_2 within VMC (having first verified the energies of the individual geminals).

Vladimir_Konjkov wrote:if changes were made to a separate git branch it would be much easier to test, without breaking master branch.

CASINO only uses a master branch; this is out of my control. In any case, nothing has been broken in the master branch (my commits pass the autotest suite).

Best,
Pablo
Hey there! I am using CASINO.
Pablo_Lopez_Rios

Posts: 50
Joined: Thu Jan 30, 2014 1:25 am

### Re: Multi-geminal wave functions

Vladimir_Konjkov wrote:I will be very grateful if you describe the algorithm for constructing a wave function (GEMINAL section) for two orthogonal geminal.

I have no experience with constructing multigeminal wave functions. I would assume that https://www.tcm.phy.cam.ac.uk/~pl275/download/pob_thesis.pdf might shed some light on this (for the specific case of the electron gas)?

Best,
Pablo
Hey there! I am using CASINO.
Pablo_Lopez_Rios

Posts: 50
Joined: Thu Jan 30, 2014 1:25 am

### Re: Multi-geminal wave functions

Vladimir_Konjkov wrote:Li-atom in cc-pVDZ basis, WFN consists of 1 geminal orbitals and an orbital filled with an unpaired electron

Unpaired electrons are indeed not supported; the geminals are supposed to be determinants of square matrices of size Nup x Ndown which must therefore satisfy Nup = Ndown. If you could point at a description of how geminals are defined for systems with Nup /= Ndown I might be able to implement this.

QCHEM uses antisymmetrized product of strongly orthogonal geminals (APSG) ansatz introdused by Rassolov in https://aip.scitation.org/doi/10.1063/1.1503773

To summarize, the complete specification of the APSSG (or SSG) model for the system of n_alpha electrons with spin up and n_beta electrons with the spin down (we assume n_alpha > n_beta ) is as follows:
The wave function has the form (5):

psi SSG = A[geminal_1(r1, r2) ... geminal_n_beta(r2n_beta-1, r2n_beta) * phi_i(r2n_beta+1)...phi_j(rn_beta + n_alpha)]

where A is an operator that antisymmetrizes the product in square brackets with respect to all electron permutations i.e. determinant.
The difference from your case is that singly occupied orbitals included as ordinary single-electron spin-orbitals - phi().

Geminals in turn can be represented as:

geminal(r1, r2) = sum Dij * A[phi(r1)phi(r2)]

where phi is spin-orbitals and i < j restriction is introduced in order to prevent the double-counting of configurations.
This equation can be re-written with the spin and spatial functions separated for a singlet geminal in the MO basis:

geminal(r1, r2) = sum Dij * F_i(r1)*F_j(r2)*(alpha(s1)*beta(s2)-beta(s2)*alpha(s1))

Spin function is antisymmetric, and so the coefficients Dij must be symmetric in order to preserve the whole wavefunction’s antisymmetry. This allows the spatial function to be reduced to diagonal form.

geminal(r1, r2) = sum Dk * PNO_k(r1)*PNO_k(r2)*(alpha(s1)*beta(s2)-beta(s2)*alpha(s1))

where PNO is (pseudo)-natural orbitals and k belongs to Arai subspaces - unintersecting supspaces of MOs which enforce strong orthogonality of the geminals.

Imposing the strong orthogonality constraint means that the one-electron space spanned by PNO is factored into n disjoint subspaces, such that each geminal is expanded in the one-electron functions belonging to that geminal’s subspace.
These subspaces are termed Arai’s subspaces, after Arai’s theorem which outlines this concept. In order for the wavefunction to be adapted to open-shell systems, the geminals are multiplied by one electron orbitals;
The one-electron orbitals are optimized under a further orthogonalization constraint;

As far as I understand QCHEM provides PNOs orbitals in MOLDEN file and Dk for every geminals as a list in output, for Be example 1-st geminal consists of 1, 12, 13, 14 orbital and 2-nd geminal consist of from 2 to 11 orbitals, I copied D weights into GEMINAL section of casl file from QCHEM output, that is, in the calculation without JASTROW factor I have to get the same energy as QCHEM output. I could not find weight of each geminal but I think they are equal to one.

Pablo_Lopez_Rios wrote:I would assume that https://www.tcm.phy.cam.ac.uk/~pl275/do ... thesis.pdf might shed some light on this (for the specific case of the electron gas)?

thanks, I started to read, it is interesting, I didn’t know that there is a connection between geminals and the BCS theory, can they be used to describe Mott insulators?

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