Orthonormal orbitals

Kohn and Sham wrote the density p(r) of the system as the sum of the square moduli of a set of one-electron orthonormal orbitals ... 

Express the three resultant orthonormal orbitals as linear eombinations of these three normalized STO s. 

In these equations, (24)-(26), orthonormal orbits are denoted by indices Vs. Equation (26) means that the orbiting electron interacting with itself, that is self-interaction, exists. This is unphysical. In order to remove this unphysical term, the SIC is taken into account by the following procedure. The SIC for the LDA in the density functional method has been treated for free atoms and insulators [16], and found an important role in determining the energy levels of electrons. However, no established formula is known to take into account the SIC for semiconductors and metals. As a way of trial, in the present calculation, the atomic SIC potential is introduced for each angular momentum in a way similar to the SIC potential for atoms [17] as follows ... 

As noted earlier, we limit ourselves arbitrarily, but judiciously, to orthonormal orbital sets in this function space, which implies the orthogonality conditions of Eq. (6). This equation represents 1/2 N(N + l)/2 conditions for the N2 matrix elements of T. Thus an orthogonal transformation of degree N contains N(N -1)/2 arbitrary parameters. Hence there exist N(N -1)/2 de-... 

N Orthonormal Orbitals Kohn-Sham-Like Equations. 207... 

Here, the minimization is over all sets of orthonormal orbitals, subject to the requirement that satisfies the (g, K) conditions. Because an ensemble of Slater determinants is incapable of describing electron correlation, one must... 

J. E. Harriman, Orthonormal orbitals for the representation of an arbitrary density. Phys. Rev. A 24,... 

Most Cl calculations involve configurations formed from a common set of orthonormal orbitals by spin and symmetry adaptation of Slater determinants. In this case S is a unit matrix and the formation of H is greatly simplified. 

For real symmetric matrices of dimension n, a triangular pattern (referred to as T) is used with the location of i,j computed as L 3 i+j(j-l)/2 for ij]. The Cl hamiltonian matrix is a large real symmetric matrix with mostly zero entries (provided orthonormal configurations constructed from orthonormal orbitals are used). If more than half the entries are zero it is more efficient to omit zero entries and include the index as a label (if the word length is long and the matrix is small enough, this label may be packed into the insignificant bits of the matrix element). 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

Probably the best-known approach to the utilization of spin symmetry is that originally developed by Slater and by Fock (see, for example, Hurley [17]). No particular advantage is taken of the spin-independence of the Hamiltonian, at least in the first phase of the construction of the n-particle basis. We take the 1-particle basis to be spin-orbitals — products of orthonormal orbitals (r) and the elementary or-thomormal functions of the spin coordinate a... 

The averaging of SCF energy expressions to impose symmetry and equivalence restrictions is a straightforward, if sometimes tedious, application of the Slater-Condon rules for matrix elements between determinants of orthonormal orbitals. This matter is discussed in detail elsewhere. The most general SCF programs can handle energy expressions of the form... 

It is straightforward also to include a core of doubly-occupied orthonormal orbitals, which may either be taken unchanged from prior calculations or optimized, simultaneously with the cip and csJ coefficients, as linear combinations of the %p. Multiconfiguration variants of the SC wavefunction may also be generated, if required, and calculations may be performed directly for excited states. 

The element (p apaq asar]v) may be written as a vacuum expectation value of a string of creation and annihilation operators such as vacuum expectation values can be expressed as a sum over all totally contracted terms, each of which depends only on the overlap between the orbitals. Since these are 5pq for orthonormal orbitals, we see that the vacuum... 

Using Slater s rules for the expansion of determinants composed of orthonormal orbitals the matrix elements of Eq. (4-6) can be reduced to combinations of orbital matrix elements of one- and two-electron operators. We can write in the usual way... 

It is written in his nearly impossible style he invented his own language for many of the standard items in the subject, such as detors , which are determinants of orthonormal orbitals . He introduces Definitions and gives Theorems, an approach which was not attractive to the students who attended his Mathematics Part III course The Quantum Theory of Molecules in Cambridge. 

The electron density is generally represented as a sum of squares of occupied orthonormal orbitals called Kohn-Sham orbitals. The use of these orbitals leads to a set of equations that are solved iteratively. The orbitals are defined with a linear combination of basis functions, comparable to Eq. [4]. Note that the Kohn-Sham orbitals are not the DFT analogy of the HE orbitals / although the basis sets for both may be of the same functional form. The Kohn-Sham orbitals are a functional representation of the electron density, and they therefore cannot be viewed as effective orbitals. 

Further use of the anticommutation relation of fermions for orthonormal orbitals gives... 

The following restrictions are hereafter imposed on the two-electron wavefunction variations. The CSF expansion coefficients C are assumed to be real and any transformation applied to these coeffieients must be real. The orbital basis is allowed to be complex but any transformation applied to the orbitals must be real. These restrictions have no effect on the expectation values of real Hamiltonian operators. Finally, an orthonormal orbital basis is assumed and only orthogonal orbital transformations are allowed. This of course does place restrictions on some of the present discussions but is considered crucial for the extension of these results to the general N-electron case. 

Just as in the two-electron case of the full Cl expansion, the simultaneous effect of both the orbital variations and the CSF expansion coefficient variations is redundant. An arbitrary wavefunction variation may be expressed by considering only the variations of the expansion coefficients for some fixed set of orthonormal orbitals. The transformation of Eq. (235) is equivalent to the... 

Even for the optimized MCSCF wavefunction, these derivatives are not zero because the energy functional W is valid only for orthonormal orbitals, and transformation (26) violates orbital orthonormality. The energy must be stationary against an infinitesimal 2x2 rotation of the form... 

Perdew and Zunger (1981), in the Xa-like equivalent of the Hartree approximation, advocate subtracting the total self-interaction of each electron in Xa-like models. This proposal would remove the m dependence of hydrogenic systems. Since the self-interaction of each electron (orthonormal orbital), as well as their sum, is not invariant under a unitary transformation among the orbitals, in contrast to the first-order density matrix and thus Xa-like models, Perdew and Zunger propose picking out a unitary transformation... 

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