Eigenvalue natural orbital 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. 

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

A way of overcoming this problem, is to use the concept of natural orbitals. The natural orbitals are those which diagonalize the density matrix, and the eigenvalues are the occupation numbers. Orbitals with occupation numbers significantly different from 0 or 2 (for a closed-shell system) are usually those which are the most important to include in tlie active space. An RHF wave function will have occupation numbers of exactly 0 or 2, and some electron correlation must be included to obtain orbitals with non-integer occupation numbers. This may for... 

The orbital occupation numbers n (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the (spin) orbital. Representing the exact density will require an infinite number of natural orbitals, with the Aieiec first having occupation numbers close to 1, and the remaining close to 0. Since the exact density matrix is not known, an (approximate) density can be written in terms of a set of auxiliary one-electron functions, i.e. orbitals. 

The Xk will also be natural orbitals but if is degenerate, they may be a unitary transformation of the Xk corresponding to the same eigenvalue n. For example, the simplest two-electron wave function for the helium atom is... 

This shows that p is similar to the well-known one-particle density of electronic structure theory. Diagonalizing the operator p yields the natural populations and the natural orbitals" defined as the eigenvalues and eigenvectors of p(p). Since we are dealing with distinguishable particles we have a separate density matrix for each degree of freedom. The natural populations characterize the contribution of the related natural orbital to the MCTDH wave function. Small natural populations therefore indicate that the MCTDH expansion converges. The natural populations thus provide us with... 

Secondly, we note that the density matrix may become singular. is a Hermitian, positive semidefinite matrix and - as mentioned above - its eigenvalues, called natural weights, characterize the importance of the corresponding natural orbital. A zero eigenvalue occurs if there is a natural orbital (i.e., a linear combination of the single-particle functions) that does not contribute to the MCTDH wave function. Its time evolution may thus be modified by replacing with p( )... 

A choice of basis set implies a partitioning of the Hamiltonian, H = Hel -I- Hso + Tn(R) + Hrot, into two parts a part, H ° which is fully diagonal in the selected basis set, and a residual part, H(1b The basis sets associated with the various Hund s cases reflect different choices of the parts of H that are included in fP°). Although in principle the eigenvalues of H are unaffected by the choice of basis, as long as this basis set forms a complete set of functions, one basis set is usually more convenient to use or better suited than the others for a particular problem. Convenience is a function of both the nature of the computational method and the relative sizes of electronic, spin-orbit, vibrational, and rotational energies. The angular momentum basis sets, from which Hund s cases (a)-(e) bases derive, are... 

Note that the eigenvalues of this matrix are just one minus the natural occupations of the target s ground state. If the target does contain an uncorrelated hard core, the matrix cannot be inverted in the full space of single-particle indices. Rather one has to exclude the hard core explicitly from the accessible space for the projectile s wave function. In the important special case where the target wave function is a Slater determinant, the matrix 2 becomes a projector onto the virtual orbital space. 

The function Xi is called a natural spin orbital (NSO). The eigenvalue v, is the occupation number of Xi- It may be shown that the sum of occupation numbers is equal to N. There is generally an infinite set of NSOs, except when the wave function is approximated by a finite number of Slater determinants. 

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