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Interpretation of exchange and correlation using natural orbital CI expansions

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Abstract

In this work we propose definitions of exchange and correlation at post-SCF level within the framework of the natural orbital CI expansion of the wave function. First, following the assumption that the Coulomb correlation is introduced through the crossed products of Slater determinants, we define exchange and correlation densities having a simple mathematical and physical interpretation and whose associated holes satisfy a series of necessary constrains and normalization rules. Thus, in our scheme the exchange and correlation densities are associated with intra- and inter-configurational terms, respectively. In turn, two different terms contribute to the correlation density. One stems from the fluctuations experienced by the 1-electron density among different electron configurations. A second term corresponds to the inter-configurational part of the 2-electron density. Expressions for post-SCF exchange and correlation potentials and bond orders have been obtained and implemented for the case of two electrons.

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Notes

  1. Although FCI calculations are unmanageable even for systems with few electrons and medium basis sets, it is better to workout our mathematical treatment from a FCI wave function, avoiding the problem of the absence of “orbital energies” for NSOs. In practical applications, the electron configurations in iterative natural orbital CI are selected by looking at the occupation numbers (instead of the orbital energies) of the NSOs resulting from the diagonalization of the 1-electron density matrix in the previous step.

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Acknowledgments

The author thanks Xunta de Galicia from the Galician government and Ministerio de Ciencia y Tecnología from the Spanish government for financial support. Special thanks to Prof. Christian Van Alsenoy for carefully reading the manuscript and for giving helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Physical Chemistry, University of Vigo, Lagoas-Marcosende s/n, 36310, Vigo, Spain

    Marcos Mandado

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  1. Marcos Mandado
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Correspondence to Marcos Mandado.

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Appendix

Appendix

Starting from Eqs. (6) and (7), in which the αα and αβ 2-electron densities are expressed as summation of intra-configurational and inter-configurational parts, and replacing each intra-configurational term by the expression of the 2-electron density for mono-configurational wave functions as given by:

$$ \rho_{II}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right) = \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\alpha } \left( {r_{2} } \right) - \rho_{X,I}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right) $$
(45)
$$ \rho_{II}^{\alpha \beta } \left( {r_{1} ,r_{2} } \right) = \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\beta } \left( {r_{2} } \right) $$
(46)

one obtains

$$ \rho^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right) = \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\alpha } \left( {r_{2} } \right) - \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{X,I}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right)} + \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I} c_{J} \rho_{IJ}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right)} } } $$
(47)
$$ \rho^{\alpha \beta } \left( {r_{1} ,r_{2} } \right) = \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\beta } \left( {r_{2} } \right) + \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I} c_{J} \rho_{IJ}^{\alpha \beta } \left( {r_{1} ,r_{2} } \right)} } } $$
(48)

where \( \rho_{X,I}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right) \) represents the exchange density for a mono-configurational wave function. On the other hand, the αα and αβ products of 1-electron densities are given by:

$$ \rho^{\alpha } \left( {r_{1} } \right)\rho^{\alpha } \left( {r_{2} } \right) = \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J = 1}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\alpha } \left( {r_{2} } \right)} } $$
(49)
$$ \rho^{\alpha } \left( {r_{1} } \right)\rho^{\beta } \left( {r_{2} } \right) = \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J = 1}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\beta } \left( {r_{2} } \right)} } $$
(50)

Introducing Eqs. (47) and (49) into Eq. (10) (with σ = α and σ′ = α), and Eqs. (48) and (50) into Eq. (10) (with σ = α and σ′ = β) one arrives at:

$$ \begin{aligned} \rho_{\text{XC}}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right) & = \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J = 1}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\alpha } \left( {r_{2} } \right)} } - \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\alpha } \left( {r_{2} } \right)} \\ & \quad + \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{X,I}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right)} - \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I} c_{J} \rho_{IJ}^{\alpha \alpha } \left( {r_{1} ,r_{2} } \right)} } \\ \end{aligned} $$
(51)
$$ \rho_{\text{XC}}^{\alpha \beta } \left( {r_{1} ,r_{2} } \right) = \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J = 1}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\beta } \left( {r_{2} } \right)} } - \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\beta } \left( {r_{2} } \right) - \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I} c_{J} \rho_{IJ}^{\alpha \beta } \left( {r_{1} ,r_{2} } \right)} } } $$
(52)

The first two terms on the right hand of Eqs. (51) and (52) can be combined yielding, respectively, Eqs. (53) and (54),

$$ \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\alpha } \left( {r_{2} } \right)} + \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} (c_{I}^{2} - 1)\rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\alpha } \left( {r_{2} } \right)} } $$
(53)
$$ \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{JJ}^{\beta } \left( {r_{2} } \right)} + \sum\limits_{I = 1}^{\text{ND}} {c_{I}^{2} (c_{I}^{2} - 1)\rho_{II}^{\alpha } \left( {r_{1} } \right)\rho_{II}^{\beta } \left( {r_{2} } \right)} } $$
(54)

which then yield, taking into account that \( (c_{I}^{2} - 1) = - \sum\nolimits_{J \ne I} {c_{J}^{2} } \),

$$ \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\left[ {\rho_{JJ}^{\alpha } \left( {r_{2} } \right) - \rho_{II}^{\alpha } \left( {r_{2} } \right)} \right]} } $$
(55)
$$ \sum\limits_{I = 1}^{\text{ND}} {\sum\limits_{J \ne I}^{\text{ND}} {c_{I}^{2} c_{J}^{2} \rho_{II}^{\alpha } \left( {r_{1} } \right)\left[ {\rho_{JJ}^{\beta } \left( {r_{2} } \right) - \rho_{II}^{\beta } \left( {r_{2} } \right)} \right]} } $$
(56)

Replacing Eqs. (55) and (56) in Eqs. (51) and (52) finally leads to Eqs. (11) and (12), respectively.

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Mandado, M. Interpretation of exchange and correlation using natural orbital CI expansions. Theor Chem Acc 132, 1364 (2013). https://doi.org/10.1007/s00214-013-1364-6

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