Orthonormality constraints

Off-diagonal elements of P corresponding to non-bonded atoms can be negative. They arise because P has to satisfy the orthonormality constraint PSP = 4P. They are not assigned any deep physical meaning. 

The AB supermolecule is described by a single determinant wave function formulated in terms of doubly occupied molecular orbitals with no orthonormality constraints. For a system with 2N = 2Na +2Nb electrons the SCF-MI wave function expressed in terms of the antisymmetrizer operator A is... 

More fundamentally, what Pecora seems to assume - although never explicitly saying so - is the following property. Since the condition CC+ = In is actually the orthonormalization constraint on the scalar product between any two wavefunctions (ft is hermitian. That is to say, it is assumed that the subspace on which the projection is made is a Hilbert subspace. 

Next we impose the orthonormality constraint on the wave functions by means of Lagrange multipliers, sy, and obtain the n one-electron Euler-Lagrange equations ... 

Associate the Lagrange multiplier ji (chemical potential) with the normalization condition in Eq. (6), the set of Hermitian-Lagrange multipliers X[ with orthonormality constraints in Eq. (4), and define the auxiliary functional Q, by the formula... 

The best NSOs are those that minimize the electronic energy subject to orthonormality constraints (4), and hence satisfy Lowdin s Eqs. (45) and (46). For the energy functional, Eq. (91), these equations become the spatial orbital Euler... 

Here the L" and R" renormalization tensors are uniquely specified by the additional orthonormality constraints... 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

When the orbitals are determined in this manner, with the only restriction being the orthonormality constraint, equation (12), they yield the best possible antisymmetrized product wavefunction (i.e., a single determinant wavefunction) for the system in question since the resulting fP(x R) yields the lowest possible value for E(R). The heart of the Hartree-Fock model is to replace the detailed and accurate description of the repulsions between every pair of electrons in the system by the average field that each electron exerts on every other. This is a consequence of the one-electron or orbital basis of the model which leads to a product wavefunction. The probability, Pa(ri,r2)dridr2, that electron 1 be in some small volume element about ri when electron 2 is simultaneously in some small volume element about ra is, in a simple product-type wavefunction, given by the product of the two singleparticle probabilities,... 

The generality and importance of the above results cannot be overemphasized. The wavefunction for the valence electrons may be optimized by variation of y alone, using the effective Hamiltonian in (47) with appropriate orthonormality constraints. In practice this means that, for a function built up from orbitals r of the valence space, it is only necessary to replace matrix elements < r h a > by... 

Equations (82) and (83) are coupled and have to be solved simultaneously. This scheme has been applied rather successfully to d cribe the melting of bulk sodium [38]. Compared to the Car-Parrinello method [44-46] the scheme has the advantage of not requiring the imposition of orthonormality constraints in the electronic equations of motion. 

The objective is to minimize the total energy as a function of the molecular orbitals, subject to the orthonormality constraint. In the above formulation this is handled bjf ... 

In order to perform this minimization under the orthonormality constraints, we use the Lagrange multiplier technique, thus minimizing... 

In this last expression, /j, is an additional, non-physical parameter, which represents the fictitious mass assigned to the additional degrees of freedom, Cj(G) s, of the system. The potential energy of the system as a whole is (c, R) = ( j(c),R), the electron+nuclei total energy functional in the Khon-Sham framework. Finally, Aij are Lagrange multipliers introduced to satisfy at all times the orthonormality constraints of the Kohn-Sham orbitals. 

More fundamentally, what Pecora seems to assume - although never explicitly saying so - is the following property. Since the condition QQ+ = In is actually the orthonormalization constraint on the since QQ = it is supposed... 

So that we may combine this expression with the orthonormality constraint eqn ( 2.4) to give... 

The orbital form of the trial density ensures that it is n-representable and so we seek a minimum in W[p x) subject to this orthonormality constraint on the orbitals Xi(x). Since the functionals T, V and J are ejl avaiilable explicitly in terms of the orbitals, the variational problem becomes identical to the Hartree-Fock variational problem set up and solved in Chapter 2 except for the problematic exchange-correlation functional Exc which is not known explicitly as a functional of p x) or the orbitals Xt(x). Thus we must simply carry the variation in Exc induced by a variation in p(x) into the differential equation for the optimum orbitals... 

Combining the energy variation and the orthonormality constraint in the usual way and insisting on the vanishing of the resulting expression for arbitrary variations and SC generates twice as many equations as there are shells ... 

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