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A new density functional theory approach based on a complete active space self-consistent field (CASSCF) reference function in Extended Koopmans’ approximation is discussed. Recently, the number of generalizations of density functional theory based on a multiconfigurational CASSCF reference function with exact exchange (CASDFT) was introduced. It was shown by one of the authors (Dr. Gusarov) that such a theory could be formulated by introducing a special form of exchange-correlation potential. To take into account an active space and to avoid double counting of correlation energy the dependence from on-top pair density
P
_{2}(
r) as a new variable was introduced. Unfortunately, this requires a deep review and reparametrization of existing functional expressions which lead to additional computational difficulties. The presented approach does not require introducing additional variables (like on-top pair density,
P
_{2}(
r)) and is based on Extended Koopmans’ theorem (EKT) approximation for multiconfigurational wave function within CASSCF method.

The recent success of computational chemistry in different areas of bio- and material sciences owes much to the development of density functional theory (DFT) methods [

For each CASSCF active space M we can construct a Green Function in the Extended Koopmans’ approximation (G^{EKT}^{,M}) by solving the generalized eigenvalue problem for the Koopmans’ matrix (see [^{EKT}^{,M} is a much simpler object in comparison with the full one, because it has a simpler pole structure [^{CASSCF}^{,M}) and electron density (ρ^{CASSCF}^{,M}) for selected active space M. Using G^{EKT}^{,M} and Dyson equation we can uniquely decompose E^{CASSCF}^{,M} into perturbation theory (PT) like series (see [

E C A S S C F , M = E C A S S C F , M ( 0 ) + E C A S S C F , M ( 1 ) + E C A S S C F , M ( 2 ) + ⋯ (1)

Of course, if M 1 ⊂ M 2 ⊂ M 3 ⊂ ⋯ ⊂ F C I then E 1 ≥ E 2 ≥ E 3 ≥ ⋯ ≥ E F C I due to variational character of CASSCF approach but the hope is that (and this is an approximation) starting from some reasonable M_{k}:

E C A S S C F , M k ( 0 ) ≅ E C A S S C F , M l ( 0 ) , M k ⊂ M l (2)

If this is true, we can define the new universal auxiliary functional:

F ˜ C A S S C F [ ρ ] = lim M ˜ → F C I ( E C A S S C F , M ˜ − E C A S S C F , M ˜ ( 0 ) ) , (3)

and then use it to construct the new CASDFT functional:

F C A S S C F , M [ ρ ] = F ˜ C A S S C F [ ρ ] − ( E C A S S C F , M − E C A S S C F , M ( 0 ) ) , (4)

which depends on active space M. The expression in the parenthesis of (4) represents the part of correlation energy already accounted by CASSCF with active space M and so should be subtracted from F ˜ C A S S C F [ ρ ] to avoid double counting.

The above considerations can be summarized by the following algorithm:

1) Suppose we have the solution of CASSCF problem for reasonably small active space M which correctly represents an electronic structure. This resulted in the WF satisfying the BLB conditions, Ψ C A S S C F , M ( r ) as well as density, ρ C A S S C F , M ( r ) and, CASSCF energy, E C A S S C F , M ;

2) Based on that solution, we can solve a generalized eigenvalue problem for Koopmans’ matrix and then construct one-particle Green’s function in Extended Koopmans’ approximation G E K T , M [

3) Next, we can decompose E C A S S C F , M into perturbation-like series [

4) Check for convergence if the termination conditions are satisfied and go to first step (if needed).

The functional F C A S S C F , M [ ρ ] in (4) differs from traditional exchange-correlation F x c [ ρ ] in Kohn-Sham (KS) approach [

lim M ˜ → F C I F C A S S C F , M ˜ [ ρ ] → 0. (5)

Based on the above, we further believe that EKT-CASDFT approach will be free of some sickness of traditional DFT. Moreover, the majority of existing DFT functionals could be easily reparametrized to approximate F ˜ C A S S C F [ ρ ] because it does not depend on any additional variables (e.g. P_{2}) which significantly simplifies its practical implementation.

The developed approach extends the DFT theory to multiconfigurational wave function which allows better accounting both static and dynamic correlation energy. Double counting of correlation energy is avoided by subtraction of the correlation energy accounted by CASSCF from universal functional F ˜ C A S S C F [ ρ ] . Details of construction and practical algorithm for implementation are presented. The practical aspects of implementation such as accuracy, convergence and computational cost are going to be studied in future works.

We would like to thank Dr. Andriy Kovalenko (NRC) and Prof. Per-Ake Malmquist (Lund University) for the insightful discussions.

Gusarov, S. and Dmitriev, Y. (2018) Extended Koopmans’ Approximation for CASDFT Exchange- Correlation Functional. Journal of Applied Mathematics and Physics, 6, 1242-1246. https://doi.org/10.4236/jamp.2018.66104