Two electron integrals

Finally, the two particle matrix elements between two ionic channel functions look as:

1

NhIα|ATˆA|Jβi

= 1

2hi1i2|t|j1j2iρ^IJ_i1i2j1j2hα|βi+hiα|t|jβiρ^IJ_ij − hiα|t|βjiρ^IJ_ij

− hi1|βihαi2|t|j1j2iρ^IJ_i1i2j1j2 − hi1i2|t|βj2ihα|j1iρ^IJ_i1i2j1j2

− 1

2hi1|βihi2i3|t|j2j3ihα|j1iρ^IJ_i1i2i3j1j2j3.

(5.18)

Note: The current treatment needs upto three particle reduced density matrices. The non-orthogonality of the single electron basis, α, with respect to the Hartree-Fock (HF) basis, φk, leads to a significant complexity in the matrix elements. Explicit orthogonalization of the single electron basis with respect to HF orbitals is not a solution because: the ionic and neutral states are treated on the configuration interaction level and each ionic state is composed of several Slater determinants.

be expressed on a quadrature grid as:

α(~r) = X

q

αqYlαmα. (5.21)

The largest angular momenta in this one-electron expansion is denoted using symbolsl_max and mmax.

There are four types of two-electron integrals that appear in the haCC scheme: hαφi|V^{2}|βφjiρ^IJ_ij , hαφi|V^{2}|φjβiρ^IJ_ij , hαφm|V^{2}|φlφniρ^IJ_klmn and hαφl|V^{2}|φmφkiη_klm^IN (See section (5.1.3)).

Outline of the steps following to compute them in the current implementation is presented in the following subsections. There exists a trade-off between the storage requirements and the operations count while designing a suitable algorithm.

5.2.1 Hartree term

The Hartree term: hαφ_i|V^{2}|βφ_jiρ^IJ_ij , is the easiest of the all the four varieties as the integral over the molecular orbitals yields an effective potential for the second electron coordinate. Denoting the matrix element as M_αβ^IJ:

M_αβ^IJ =X

ij

hαφi|V^{2}|βφjiρ^IJ_ij

=X

ij

ρ^IJ_ij X

qq⁰

X

LM

4π

2L+ 1α^∗_qβ_qV_qq^L0

X

limi

c^i∗_q0limi

X

ljmj

c^j_q0ljmjhY_lαmαY_LM|Y_lβmβihY_limi|Y_LMY_ljmji (5.22) the integral evaluation is performed using the following steps:

1. First, an effective potential object R_{LM q}^IJ is defined as follows:

R^IJ_{LM q} = 4π 2L+ 1

X

ij

ρ^IJ_ij X

q⁰

V_qq^L0

X

limi

c^i∗_q0l_im_i

X

hjmj

c^j_q0ljmjhYlimi|YLMYljmji (5.23)

which is computed at the start and stored. The next steps are done on the fly.

2. For each I, J, α, β the vector Tq is computed as:

Tq =X

LM

R^IJ_{LM q}hYlαmαYLM|Ylβmβi (5.24)

3. This yields the required integral

M_αβ^IJ =X

q

α^∗_qT_qβ_q (5.25)

The limits of LM expansion define the required storage and this can be obtained by examining equations 5.23 and 5.24. The needed LM are:

Lmin = 0

Lmax = 2 min(lmax, l_max^g )

Mmin(L) = max(max(−2m^g_max,−2mmax),−L) Mmax(L) = min(min(2m^g_max,2mmax), L)

5.2.2 Standard exchange term

The standard exchange term which is the most expensive of all can be expressed as:

M_αβ^IJ =X

ij

hαφi|V^{2}|φjβiρ^IJ_ij

=X

ij

ρ^IJ_ij X

qq⁰

X

LM

4π

2L+ 1α^∗_qβq⁰V_qq^L0

X

limi

c^i∗_q0limihYlimiYLM|YlβmβiX

ljmj

c^j_qljmjhYlαmα|YLMYljmji (5.26) which can be recast as:

M_αβ^IJ =X

qq⁰

α^∗_qβq

X

LM

4π 2L+ 1V_qq^L0

X

ij

R^LM_αjqρ^IJ_ij R^†LM_βiq0 (5.27) where

R_αjq^LM =X

ljmj

c^j_qljmjhYlαmα|YLMYljmji (5.28) The expression within the ij summation can be simplified by performing a singular value decomposition or in other words by transforming from the molecular orbital basis to natural orbital basis. In the natural orbital basis ρ^IJ_ij is a diagonal matrix. Using the singular value decompositionρ^IJ_ji =U_jx^IJV_xi^IJ†,

X

ij

R^LM_αjqρ^IJ_ij R^†LM_βiq0 =X

ij

R^LM_αjqU_jx^IJV_xi^IJ†R^†LM_βiq0 =X

x

T_αxq^IJLMT_βxq^†IJLM0 (5.29) This reduces the number of required floating point operations when the number of natural orbitals is smaller than the number of Hartree-Fock orbitals. The reduction in the double summation ij to single summation is compensated by the fact that T is also a function of IJ unlike R.

The original expression for the exchange integral can be re-written as:

M_αβ^IJ =X

qq⁰

α_q^∗β_qX

LM

4π 2L+ 1V_qq^L0

X

x

T_αxq^IJLMT_βxq^†IJLM₀ (5.30) The integrals are computed using the following steps:

1. The objects T are computed and stored. This is possible for atomic case. In the molecular case due to the large angular momentum requirements, storing these may not be possible.

2. The next steps are done on the fly. An object S_qq^IJ0αβ is evaluated as:

S_qq^IJ0αβ =X

LM

4π 2L+ 1V_qq^L0

X

x

T_αxq^IJLMT_βxq^†IJLM0 (5.31)

3. This leads to the required integral:

M_αβ^IJ =X

qq⁰

α^∗_qS_qq^IJ0αββq (5.32) The multipole expansion truncations can be obtained by examining the integrals between the spherical harmonics. In the current case the LM truncation limits are:

Lmin = max(0, la−l^g_max) Lmax = la+l^g_max

Mmin = min(max(−L, ma−m^g_max), L) Mmax = max(min(L, ma+m^g_max),−L)

5.2.3 Non-standard two-electron integral: h αφ

b

| V

^{2}

| φ

c

φ

d

i ρ

^IJ_abcd

There are other non-standard exchange terms that appear in the two-particle operator due to non-orthogonality of the molecular orbitals from quantum chemistry with respect to the single electron finite element basis. The integral

M_αa^IJ =X

bcd

hαφb|V^{2}|φcφdiρ^IJ_abcd

=X

bdc

ρ^IJ_abcdX

qq⁰

X

LM

4π

2L+ 1α^∗_qV_qq^L0

X

l_bm_b

c^b∗_q0l_bm_b

X

lcmc

c^c_qlcmcX

l_dm_d

c^d_q0l_dm_dhYlbmb|YLMYldmdihYlαmαYLM|Ylcmci (5.33)

is evaluated using the following steps:

1. A direct potential like object P_{LM q}^bd is first constructed:

P_{LM q}^bd = 4π 2L+ 1

X

q⁰

V_qq^L0

X

lbmb

c^b_q^∗0lbmb

X

ldmd

c^d_q0ldmdhYl_bm_b|YLMYl_dm_di (5.34) 2. For each l_α, m_α an intermediate objectQ^cq_bd

Q^cq_bd =X

lcmc

c^c_qlcmcX

LM

hYlαmαYLM|YlcmciP_{LM q}^bd (5.35)

3. Using the object Q, an object T_qa^IJ is constructed as:

T_qa^IJ =X

bcd

Q^cq_bdρ^IJ_abcd (5.36)

which is computed and stored.

4. This leads to the required result:

M_αa^IJ =X

q

T_qa^IJα_q^∗ (5.37)

The multipole expansions are truncated as:

Lmin = 0

L_max = min(2l_max^g , l_max+l_max^g )

Mmin = max(−L,−2m^g_max,−mmax−m^g_max) Mmax = min(L,2m^g_max, mmax+m^g_max)

5.2.4 Non-standard two-electron integral: h φ

a

φ

b

| V

^{2}

| φ

d

β i η

^N_abd^J

The steps followed for this integral is very similar to the previous one. The integral can be expanded using the multi-pole expansion as:

M_β^NJ =X

abd

hφaφb|V^{2}|φdβiη_abd^NJ

=X

abd

η_abd^NJX

qq⁰

X

LM

4π

2L+ 1β_q0V_qq^L0

X

lama

c^a∗_qlamaX

lbmb

c^b∗_q0l_bm_b

X

ldmd

c^d_qldmdhY_lama|Y_LMY_ldmdihY_lbmbY_LM|Y_lβmβi (5.38)

which is evaluated using the following steps.

1. Construct the potential P_{LM q}^ad 0

P_{LM q}^ad 0 = 4π 2L+ 1

X

q

V_qq^L0

X

lama

c^a_ql^∗_ama X

ldmd

c^d_qldmdhYlama|YLMYldmdi (5.39)

2. For each lβ and mβ construct an intermediate object Q^bq_ab⁰as:

Q^bq_ab⁰ =X

lbmb

c^b_q^∗0lbmb

X

LM

hYl_bm_bYLM|Yl_βm_βiP_lmq^ad 0 (5.40)

3. Construct object Tq⁰ as:

T_q^N0 ^J =X

abd

Q^bq_ad⁰η_abd^NJ (5.41) This object is computed initially for each lβ and mβ and stored.

4. Finally, the required matrix element is computed on the fly using the stored object T as:

M_β^NJ =X

q⁰

T_q^N0^Jβq⁰ (5.42)

The multi-pole expansions are truncated as:

Lmin = 0

Lmax = min(2l_max^g , lmax+l_max^g )

M_min = max(−L,−2m^g_max,−m_max−m^g_max) Mmax = min(L,2m^g_max, mmax+m^g_max)

Note: As finite elements are used, the angular momentum limits can also be made a function of the finite element number.

Im Dokument Strong field single ionization of atoms and small molecules (Seite 123-128)