Orthonormal Subject

Roothaan actually solved the problem by allowing the LCAO coefficients to vary, subject to the LCAO orbitals remaining orthonormal. He showed that the LCAO coefficients are given from the following matrix eigenvalue equation ... 

We then allow Ri and R2 to vary, subject to orthonormality, just as in the closed-shell case. Just as in the closed-shell case, Roothaan (1960) showed how to write a Hamiltonian matrix whose eigenvectors give the columns U] and U2 above. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

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 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... 

Please minimize the electronic energy of this single determinant subject to the constraint that the spin orbitals all remain orthonormal to one another. 

Not all variations of the orbital set are allowed. The variations are subject to the constraint that the orbitals remain orthonormal [equation (A.11)]. Thus, for all pairs of orbitals a and b,... 

Finally, some remarks will be made concerning the dimension of wavefunctions. The bound-state orbitals are subject to the orthonormality relation... 

As Dewar points out in ref. [30a], this derivation is not really satisfactory. A rigorous approach is a simplified version of the derivation of the Hartree-Fock equations (Chapter 5, Section 5.2.3). It starts with the total molecular wavefunction expressed as a determinant, writes the energy in terms of this wavefunction and the Hamiltonian and finds the condition for minimum energy subject to the molecular orbitals being orthonormal (cf. orthogonal matrices, Section 4.3.3). The procedure is explained in some detail in Chapter 5, Section 5.2.3)... 

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... 

These energy functionals are made variational by requiring they be stationary with respect to variation of the orbital coefficients, occupation numbers, x for analytic Xa, and d subject to the constraints that the orbitals be orthonormal and the sum of the occupation numbers be the total number of electrons, Ne, and optional constraints on the fits. 

This new Hamiltonian is with respect to an orthonormal basis. If the square-root is that of Lowdin orthogonalization [ 117,118], then the new basis is [117,119] as similar as possible (in a well-defined mathematical sense) to the initial basis subject to the constraint of orthogonality. 

In order to evaluate 5s/ISv from Eq. (282), we further need the functional derivatives dqfjjdvs and ScpflSv. The stationary OPM eigenfunctions (< /r), = 1,..., oo) form a complete orthonormal t, and so do the time-evolved states unperturbed states, remembering that at t = ti the orbitals are held fixed with respect to variations in the total potential. We therefore start from t = ti, subject the system to an additional small perturbation (5i)s(r, t) and let it evolve backward in time. The corresponding perturbed wave functions [Pg.135]

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 ... 

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. 

This is to be solved subject to the boundary condition that as p tends to zero, due to the orthonormality and completeness of the eigenfunctions tj, C tends to the delta function, i.e. 

The expansion coefficients, A/ and ), , in the sums in equations 6 and 8 are the variable parameters in the formalism. They are called the configuration interaction (Cl) coefficients and the MO coefficients respectively. The optimum values of these parameters, at a particular nuclear geometry, are found by minimising the energy expression for the system with respect to the variables, subject to the constraints that the orbitals are orthonormal ... 

The ground state p(r) which is the solution to Eq. (9) is then that which minimizes [ (r)], subject to the constraint that the one-electron orbital wavefunctions, V, s, which constitute /9(r) as expressed in Eq. (8), are orthonormal ... 

The energy and spin orbitals are then determined variationally, subject to the constraint that the spin orbitals are orthonormal. This leads to the familiar HF integro-differential equations for the "best" one-electron orbitals. Physically, the HF approximation amounts to treating the individual electrons in the average field due to all the other electrons in the system. This effective Hamiltonian is called the Fock operator. 

Upon introducing the LCAO approximation and then minimizing the total energy in Eq. (20) with respect to the MO coefficients (subject to the orthonormality of the MO s), the familiar algebraic equations for the canonical orbitals, analogous to the Pople-Nesbet equations [56] in HF theory, are obtained, with the XC matrix elements given by [57]... 

---