Geometrical constraints to reduce complexity in quantum molecular systems

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Geometrical constraints to reduce complexity in quantum molecular systems

Thomas Pérez, Patrick Cassam-Chenaï

To cite this version:

Thomas Pérez, Patrick Cassam-Chenaï. Geometrical constraints to reduce complexity in quantum molecular systems. UCA Complex days 2019, Mar 2019, Nice, France. �hal-02531863�

Geometrical constraints to reduce complexity in quantum molecular systems

Thomas Perez

, Patrick Cassam-Chenaï

1 _{Laboratoire J. A. Dieudonné, UMR 7351 UCA-CNRS, Nice (tperez@unice.fr, cassam@unice.fr)}

P

RESENTATION OF THE PROBLEM

The energy of a molecule is not the sum of the energies of its atomic components. Our aim is to compute an accurate approximation of this quantity based on an electron pair model, that is to say by using an

antisymmetric product of two-electron wave functions, called “geminals”. In this model, the total wave function ‰<sub>e D g1 ^


^ gn can also be described by a set of matrices</sub> C_{k D ci;j}k

1i;j m , one for each

geminal g_{k D} P

1i;j m

c_i;jk '_{i ^ 'j , where} .'_i /_{1im (resp.} .'_j /_{1j m) is a basis orbital of spin C} 1_{2 (resp.} 1_{2 ).}

However, without further restrictions, such a model has a factorial computational complexity with the number of electrons determined by the calculation of geminal product overlaps and its applicability is therefore limited to small systems. We will introduce generalized orthogonality constraints between gem-inals to reduce the computational effort, without sacrificing the indistinguishability of the electrons.

G

ENERAL GEMINAL PRODUCT OVERLAP FORMULA

Theorem 1 : For k _{2 f1; : : : ; ng, let g}k and g_k0 be geminals whose associated matrices are denoted respectively Ck

and C_k0 . The overlap between the wave functions ‰e D g1 ^


^ gn and ‰_e0 D g₁0 ^


^ g_n0 is given by the formula :

h‰ej‰_e0 i D hg1 ^


^ gnjg₁0 ^


^ g_n0 i D X 0Nn;0;:::;Nn;nn n P i_D0 N_{n;i D} n P i_D0 iN_n;iDn . 1/Nn;0 X ;0_2Sn n Y i D1 N_n;i Q j D1 T_NNn;i n;0;:::;Nn;i 1 .;  0_/ iNn;i N n;i Š with : T Nn;i Nn;0;:::;Nn;i 1.; 0/ D tr h CŽ  i 1 P p_D0 pNn;pC.j 1/iC1
C0  0 i 1 P p_D0 pNn;pC.j 1/iC1



CŽ  i 1 P p_D0 pNn;pCj i
C0  0 i 1 P p_D0 pNn;pCj i
i :

Example .n _{D 3/ : We compute hg}1 ^ g2 ^ g3jg₁0 ^ g₂0 ^ g₃0 i D A1 C A2 C A3 , as follow :

I 3=0+0+3 : N3;0 D 2; N3;1 D 0; N3;2 D 0; N3;3 D 1 A1 D P ;0_2S3 tr h C_{ .1/}Ž C0  0 .1/C Ž  .2/C_{ 0 .2/}0 C Ž  .3/C_{ 0 .3/}0 i 3 . I 3=0+1+2 : N3;0 D 1; N3;1 D 1; N3;2 D 1; N3;3 D 0 A2 D P ;0_2S3 tr h C_{ .1/}Ž C0  0 .1/ i tr h C_{ .2/}Ž C0  0 .2/C Ž  .3/C_{ 0 .3/}0 i 2 . I 3=1+1+1 : N3;0 D 0; N3;1 D 3; N3;2 D 0; N3;3 D 0 A3 D P ;0_2S3 tr h C_{ .1/}Ž C0  0 .1/ i tr h C_{ .2/}Ž C0  0 .2/ i tr h C_{ .3/}Ž C0  0 .3/ i 6 .

S

CALING OF THE OVERLAP FORMULA FOR

H

LINEAR CHAINS

Hm molecules H6 H8 H10 H12 H14 H16

.n; number of partitions of n/ .3; 3/ .4; 5/ .5; 7/ .6; 11/ .7; 15/ .8; 22/

Number of terms in the sum (_{D .nŠ/}2) 36 576 14 400 518 400 25 401 600 1 625 702 400

The computational cost rises too fast and makes the general model unpractical.

I

NTRODUCTION OF CONSTRAINTS FOR A SIMPLER ANSATZ

We will consider wave functions which are products of n singlet or triplet geminals (i.e. with symmetric or antisymmetric associated matrices) and we will impose to our geminals g_{k’s the so-called} permutationally invariant 2-orthogonality constraints :

8i; j; k 2 f1; : : : ; ng distinct; (

hgi jgj i D 0

g_k ê .gi ^ gj / D 0

i.e. in terms of matrices :

(

tr.C_iŽC_j / _{D 0}

C_i C_kŽC_{j C Cj} C_kŽC_{i D 0} : I The "maximal" linearly independent family of 2  2 matrices verifying these conditions is :

I_{2 D}  1 0 0 1  I _{x D}  0 1 1 0  I _{z D}  1 0 0 1  and i _{y D}  0 1 1 0  ; where _x, _{y and} _{z are the} Pauli matrices.

E

XPRESSION OF THE OVERLAP FOR A SIMPLIFIED MODEL

Model : By considering integers h0 D 0 < h1 <


< hn, each matrix Ck (associated to gk) consists of a

1-orthogonal diagonal part of size hn  hn where only the coefficients of lines and columns hk 1 C 1; : : : ; hk

are nonzero, and of blocks j_k B_kj _{of size 2  2 each proportionnal to} I_2, _x, i _{y or} _{z :}

C_{k D} 0 B B B @ D_k 1_k B_k1 : :: m_k 0 B_km0 1 C C C A with D_{k D} 0 B B B B B @

0

0

The number of 2 _{ 2 blocks is m}0 _D m hn

2 2 N.

For all j , there is no more than one nonzero j_{k for each type of block (}I₂; _x; i _y; _z).

Moreover, each C_k0 _{has the same matrix form as} C_{k (same types of blocks and zeros at the same places).}

Theorem 2 : Let g1; : : : ; gn and g₁0 ; : : : ; g_n0 be geminals verifying our constrained model conditions. We have :

hg1 ^


^ gnjg₁0 ^


^ g_n0 i D X 0j1;:::;jnm0 distinct if nonzero _gj1 1;g₁0


 j_n g_n;g_n0 ; with  j g_u;g_u0 D 8 ˆ ˆ < ˆ ˆ : hu P t Dhu 1C1 _ut 0_u t if j _{D 0} 2 j_u0_uj otherwise

S

CALING OF THE CONSTRAINED MODEL FOR

H

LINEAR CHAINS

Hm molecules H6 H8 H10 H12 H14 H16

.n; h_n; m0/ .3; 4; 1/ .4; 4; 2/ .5; 6; 2/ .6; 6; 3/ .7; 8; 3/ .8; 8; 4/

Number of terms in the sum 4 21 22 95 100 441