Orbital correlation functionals

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

There are several types of basis functions listed below. Over the past several decades, most basis sets have been optimized to describe individual atoms at the EIF level of theory. These basis sets work very well, although not optimally, for other types of calculations. The atomic natural orbital, ANO, basis sets use primitive exponents from older EIF basis sets with coefficients obtained from the natural orbitals of correlated atom calculations to give a basis that is a bit better for correlated calculations. The correlation-consistent basis sets have been completely optimized for use with correlated calculations. Compared to ANO basis sets, correlation consistent sets give a comparable accuracy with significantly fewer primitives and thus require less CPU time. 

CASSCF wave function includes only the static correlation only a small number of electrons spanning frontier orbitals are correlated between them, while... 

As discussed in Chapter 9, the VBCI method provides results that are at par with the BOVB method, the difference being that the electrons of the spectator orbitals are correlated too in the VBCI method. The wave function starts from a VBSCF wave function and augments it with subsequent local configuration interaction that can be restricted to single excitations (VBCIS level), or single and double excitations (VBCISD), or higher excitations. Here, we will consider only the VBCISD level, which is a good compromise between accuracy and cost efficiency. 

Although a Slater-determinant reference state 4> cannot describe such electronic correlation effects as the wave-function modification required by the interelec -tronic Coulomb singularity, a variationally based choice of an optimal reference state can greatly simplify the -electron formalism. 4> defines an orthonormal set of N occupied orbital functions occupation numbers = 1. While () = 1 by construction, for any full A-electron wave function T that is to be modelled by it is convenient to adjust (T T) > 1 to the unsymmetrical... 

Orbital functional theory 5.2.3 Exact correlation energy... 

Since H is specified, Eq. (5.3) defines Ec as a functional of the occupied orbitals of orbital functional derivative = v, (J), defines a nonlocal correlation potential vc in... 

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

Since the exact ground-state electronic wave function and density can only be approximated for most A-electron systems, a variational theory is needed for the practical case exemplified by an orbital functional theory. As shown in Section 5.1, any rule T 4> defines an orbital functional theory that in principle is exact for ground states. The reference state for any A-electron wave function T determines an orbital energy functional E = Eq + Ec,in which E0 = T + Eh + Ex + V is a sum of explicit orbital functionals, and If is aresidual correlation energy functional. In practice, the combination of exchange and correlation energy is approximated by an orbital functional Exc. 

For exchange-correlation energy expressed as an orbital functional,... 

These concepts, inherent in the TDHF formalism, generalize immediately to orbital functional theory, when electronic correlation energy is included in the model. Given some definition that determines a reference state for any... 

V-clcctron state T, correlation energy can be defined for any stationary state by Ec = E — / o, where Eo = ( //1) and E = ( // 4 ). Conventional normalization ) = ( ) = 1 is assumed. A formally exact functional Fc[4>] exists for stationary states, for which a mapping — F is established by the Schrodinger equation [292], Because both and p are defined by the occupied orbital functions occupation numbers nt, /i 4>, E[p and E[ (p, ] are equivalent functionals. Since E0 is an explicit orbital functional, any approximation to Ec as an orbital functional defines a TOFT theory. Because a formally exact functional Ec exists for stationary states, linear response of such a state can also be described by a formally exact TOFT theory. In nonperturbative time-dependent theory, total energy is defined only as a mean value E(t), which lies outside the range of definition of the exact orbital functional Ec [ ] for stationary states. Although this may preclude a formally exact TOFT theory, the formalism remains valid for any model based on an approximate functional Ec. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that... 

This is a well-known result that has been obtained in several different ways [8, 30, 28, 31, 32] The exact correlation potential of DFT is known [28] to fall off as —a/(2r4) for atoms with spherical N and (N — l)-electron ground states, with a being the static polarizability of the (N — l)-electron ground state. Theorem 2 provides a simple way of checking how the OEP correlation-only potential VC(T(r) falls off for a given approximate orbital functional approx[ 0. ] One only needs to determine the asymptotic decay of ucjv [Pg.43]

The short-range form of the correlation potential has been considered in DFT primarily in terms of density-gradient corrections to the LDA [6,7]. The correlation energy functional given by Eq.(7) here is not expressed explicitly as a functional of the electronic density, but is a formally exact expression which could be used to calibrate proposed functionals in particular cases where an accurate Cl expansion of a correlated ground-state wave function is known. When occupied orbital functions are expressed in a localized representation,... 

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