Addition theorems for Slater-type orbitals and their application to multicenter multielectron integrals of central and noncentral interaction potentials

Abstract

By the use of complete orthonormal sets of ψ^α-ETOs (α=1, 0, −1, −2, ...) introduced by the author, new addition theorems are derived for STOs and arbitrary central and noncentral interaction potentials (CIPs and NCIPs). The expansion coefficients in these addition theorems are expressed through the Gaunt and Gegenbauer coefficients. Using the addition theorems obtained for STOs and potentials, general formulae in terms of three-center overlap integrals are established for the multicenter t-electron integrals of CIPs and NCIPs that arise in the solution of the N-electron atomic and molecular problem (2≤t≤N) when a Hylleraas approximation in Hartree–Fock–Roothaan theory is employed. With the help of expansion formulae for translation of STOs, the three-center overlap integrals are expressed through the two-center overlap integrals. The formulae obtained are valid for arbitrary quantum numbers, screening constants and location of orbitals.

Appendix

As can be seen from Eq. (23), the multicenter t-electron integrals are expressed through the following basic three-center one-electron integrals over STOs:

$$ {\eqalign{ & S^{{aaa...acb}}_{{p_{1} p_{2} p_{3} ...p_{{t - 1}} p_{t} p_{{t + 1}} }} (\zeta _{1} \zeta _{2} \zeta _{3} ...\zeta _{{t - 1}} \zeta _{t} \zeta _{{t + 1}} ) \cr & \,\,\,\,\,\, = ({\sqrt {4\pi } })^{{t - 1}} {\int \matrix{ \chi ^{*}_{{p_{1} }} (\zeta _{1} ,\vec{r}_{a} )\chi _{{p_{2} }} (\zeta _{2} ,\vec{r}_{a} )\chi _{{p_{3} }} (\zeta _{3} ,\vec{r}_{a} )... \hfill \cr \times \chi _{{p_{{t - 1}} }} (\zeta _{{t - 1}} ,\vec{r}_{a} )\chi _{{p_{t} }} (\zeta _{t} ,\vec{r}_{c} )\chi _{{p_{{t + 1}} }} (\zeta _{{t + 1}} ,\vec{r}_{b} ){\mathop{\rm d}\nolimits} v \hfill \cr} } \cr & \,\,\,\,\,\, = ({\sqrt {4\pi } })^{{t - 1}} {\int \matrix{ R_{{n_{1} n_{2} n_{3} ...n_{{t - 1}} }} (\zeta _{1} \zeta _{2} \zeta _{3} ...\zeta _{{t - 1}} ,r_{a} )\Theta _{{l_{1} m_{1} ,l_{2} m_{2} ,l_{3} m_{3} ,...,l_{{t - 1}} m_{{t - 1}} }} \hfill \cr (\theta _{a} \varphi _{a} )\chi _{{p_{t} }} (\zeta _{t} ,\vec{r}_{c} )\chi _{{p_{{t + 1}} }} (\zeta _{{t + 1}} ,\vec{r}_{b} ){\mathop{\rm d}\nolimits} v \hfill \cr} } \cr} } $$

(25)

where

$$ {\eqalign{ & R_{{n_{1} n_{2} n_{3} ...n_{k} }} (\zeta _{1} \zeta _{2} \zeta _{3} ...\zeta _{k} ,r) = R_{{n_{1} }} (\zeta _{1} ,r)R_{{n_{2} }} (\zeta _{2} ,r)R_{{n_{3} }} (\zeta _{3} ,r)...R_{{n_{k} }} (\zeta _{k} ,r) \cr & = {\left[ {{{(2N_{k} )!} \over {(2n_{1} )!(2n_{2} )!(2n_{3} )!...(2n_{k} )!}}} \right]}^{{1/2}} \cr & \times ({\sqrt {2z_{k} } })^{{3(k - 1)}} x^{{n_{1} + 1/2}}_{1} x^{{n_{2} + 1/2}}_{2} x^{{n_{3} + 1/2}}_{3} ...x^{{n_{k} + 1/2}}_{k} R_{{N_{k} }} (z_{k} ,r) \cr} } $$

(26)

$$ {\eqalign{ & N_{k} = n_{1} + n_{2} + n_{3} + ... + n_{k} - k + 1,\;z_{k} = \xi _{1} + \xi _{2} + \xi _{3} + ... + \xi _{k} ,x_{i} = \xi _{i} /z_{k} ,\; \cr & i = 1,2,3,...,k \cr} } $$

(27)

and

$$ {\eqalign{ & \Theta _{{l_{1} m_{1} ,l_{2} m_{2} ,l_{3} m_{3} ,...,l_{k} m_{k} }} (\theta ,\varphi ) = S^{*}_{{l_{1} m_{1} }} (\theta ,\varphi )S^{{}}_{{l_{2} m_{2} }} (\theta ,\varphi )S^{{}}_{{l_{3} m_{3} }} (\theta ,\varphi )...S^{{}}_{{l_{k} m_{k} }} (\theta ,\varphi ) \cr & = {1 \over {({\sqrt {4\pi } })^{{k - 1}} }} \cr & \times {\sum\limits_{L_{2} M_{2} ,L_{3} M_{3} ,...,L_{k} M_{k} } {d^{{L_{2} M_{2} }} (l_{1} m_{1} ,l_{2} m_{2} )d^{{L_{3} M_{3} }} (L_{2} M_{2} ,l_{3} m_{3} )...d^{{L_{k} M_{k} }} (L_{{k - 1}} M_{{k - 1}} ,l_{k} m_{k} )S^{*}_{{L_{k} M_{k} }} (\theta ,\varphi )} } \cr} } $$

(28)

$$ {d^{{LM}} (lm,{l}'{m}') = (2L + 1)^{{1/2}} C^{{L{\left| M \right|}}} (lm,{l}'{m}')A^{M}_{{m{m}'}} } $$

(29)

Here, ${R_{{N_{k} }} (z_{k} ,r)\,\,\,{\rm{and}}\,\,\,S_{{L_{k} M_{k} }} (\theta ,\varphi )}$ are the radial parts of normalized STOs and the spherical harmonics, respectively. See [18] for the exact definition of the Gaunt coefficients C ^L│M│ and the quantities $A_{mm'}^M$. We notice that, Eq. (28) can easily be derived by the use of the following expansion relation for the product of two spherical harmonics both with one and the same center [18]:

$$ {S^{ * }_{{lm}} (\theta ,\varphi )S^{{}}_{{{l}'{m}'}} (\theta ,\varphi ) = {1 \over {{\sqrt {4\pi } }}}{\sum\limits_{LM}^{} {d^{{L{\left| M \right|}}} (lm,{l}'{m}')S^{ * }_{{LM}} (\theta ,\varphi )} }} $$

(30)

Inserting Eqs. (26) and (28) into the integral in Eq. (25) leads to the three-center overlap integrals

$$ {S^{{acb}}_{{nlm,{n}'{l}'{m}',{\mu }'{\nu }'{\sigma }'}} (\zeta ,{\zeta }',{z}') = {\sqrt {4\pi } }{\int {\chi ^{*}_{{nlm}} (} }\zeta ,\vec{r}_{a} )\chi ^{{}}_{{{n}'{l}'{m}'}} ({\zeta }',\vec{r}_{c} )\chi ^{{}}_{{{\mu }'{\nu }'{\sigma }'}} ({z}',\vec{r}_{b} ){\mathop{\rm d}\nolimits} v} $$

(31)

Using the expansion formulae for electron charge densities (see Eq. (19) of [15]), the three-center overlap integrals [Eq. (31)] can be expressed through the two-center overlap integrals:

$$ {\eqalign{ & S^{{acb}}_{{nlm,{n}'{l}'{m}',{\mu }'{\nu }'{\sigma }'}} (\zeta ,{\zeta }',{z}') \cr & = {\mathop {\lim }\limits_{N \to \infty } }{\sum\limits_{\mu = 1}^N {{\sum\limits_{\nu = 0}^{\mu - 1} {{\sum\limits_{\sigma = - \nu }^\nu {W^{{\alpha N}}_{{nlm,{n}'{l}'{m}',\mu \nu \sigma }} (\zeta ,{\zeta }',z;\vec{R}_{{ca}} ,0)S^{{}}_{{\mu \nu \sigma ,{\mu }'{\nu }'{\sigma }'}} (z,{z}';\vec{R}_{{ab}} )} }} }} } \cr} } $$

(32)

where $z = \zeta + \zeta ',\;S_{\mu \nu \sigma ,\mu '\nu '\sigma '}^{} (z,z';\vec R_{ab} ) \equiv S_{\mu \nu \sigma ,\mu '\nu '\sigma '}^{ab} (z,z')\,\,{\rm{and}}\,\,W^{\alpha N}$ is the two-center charge density expansion coefficient. Thus, the basic three-center one-electron integrals (Eq. 25) are determined solely from the two-center overlap integrals.

The results of calculation for the two-center two-electron integrals of central interaction potential f ₀₀₀(0, r ₂₁)=1/r ₂₁ obtained with a Pentium III 800 MHz computer (using TURBO Pascal 7.0) are shown in Table 1. The comparative values obtained from the expansions in terms of different complete orthonormal sets of ψ^α-ETOs are shown in this table. As can be seen from the table, the computation time and accuracy of the computer results for different expansion formulae obtained from ψ⁰-ETOs, ψ¹-ETOs and ψ^–1-ETOs are satisfactory. Work is in progress for the computation of multicenter multielectron integrals of central and noncentral interaction potentials over STOs based on the formulae given in this work.

Table 1. Comparison of methods of computing two-center two-electron integrals of central interaction potential f ₀₀₀(0,r ₂₁)=1/r ₂₁ obtained in the molecular coordinate system in a.u. for N ₂₁=15, θ_ba=θ_db=45°, φ_ba=φ_db=270°