Optimization wavefunction

Having examined the process of intra-orbit optimization in the previous Sections, let us turn our attention to inter-orbit jumping . This process is shown in Fig. 7 by means of steps indicated by arrows, each of which involves the transformation of a density-optimal wavefunction e C> into another wavefunction optiP%)- We denote the latter as the orbit-generating wave-function for orbit 0. ... 

Let us note, however, that in spite of the fact that the orbit jumping optimization is carried out at fixed density, the resulting wavefunction, epi[p%) = is not necessarily associated with the fixed density p ]],(r). For this reason, it is then possible to apply a local-scaling transformation to it and produce an optimized wavefunction which, at the same time, is associated with the fixed density. We denote this wavefunction by Moreover, we can... 

Optimize wavefunction once for the given start position. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

The result of this optimization process is c lc+ = 0.16. In other words, the optimal wavefunction has ionic and covalent contributions in the ratio of about 1 6. Such a dominance of covalent character is expected for a molecule such as H2. 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

The one-electron, Coulomb and exchange integrals are analogous to Eqs (9.5)-(9.7), but in terms of MO s rather than AO s. (The 4 must now contain contributions from all the nuclei in the molecule.) The optimized wavefunction of the form (11.43) involves, in principle, the solution of N simultaneous integrodifferential Hartree-Fock equations. It is much more computationally efficient to transform these into a set of N linear algebraic equations. To do this, each of the MO s is expressed in terms of a set of n basis functions ... 

These simple examples clearly show that orbital relaxation is crucial in vibronic coupling. Therefore, variationally optimized wavefunctions should be employed for vibronic coupling calculations. The frozen orbital approximation is not suitable for calculation [35]. 

In order to highlight the particular way in which the minimization process is carried out in the Levy variational principle, let us consider Fig. 1. In the inner variation of the Levy principle one searches for the optimal wavefunction in the sense that it yields the lowest energy, corresponding to a fixed density p(r) = Pfixed(r) Af. This means, that one moves along the row whose wavefunctions yield p(r) = p/ixedix) Thus, the inner variation becomes... 

The dependence of the functional F[p(r) =i°p Ml] 0n the optimal wavefunction for a given density makes it highly unlikely that this should be a universal functional . Thus, the search for such objects in the Hohenberg-Kohn-Sham version of density functional theory stands on feeble grounds. 

Let us go back to Pig. 2. There we have sketched by means of steps formed by arrows the orbit-jumping processes whereby a density-optimal" wavefunction 0 is carried into another wavefunction opt(p lt) which we define as the orbitgenerating wavefunction for orbit 0 . In order to see what is meant by opt(p lt) consider again the energy expression given by Eq. (50) and assume that it is evaluated at pi j,f Notice that the energy functional P[ p(] = is... 

As discussed in Section 3.1.1, starting from an orbit generating wavefunction W(ri, , r,v) in position space, we may compute the optimal one-particle density Polt(r) = pii) jj(r) by optimizing the energy functional S[p x) j 1] subject to a normalization condition on the density. In other words, the optimal wavefunction , rV) within the position orbit is obtained by means of a local-scaling transformation of the orbit-generating wavefunction. 

Application of a Fourier transformation to (n,, rN) produces the wavefunction (pi, , pjv) in momentum space. We have used a tilde to indicate that this Fourier-transformed wavefunction does not necessarily correspond to the optimal wavefunction (pi, , pN) within the momentum orbit The latter satisfies the extremum condition of the variational minimization of the energy functional S[tt(p) W] subject to the normalization condition / d3pir(p) = N. 

G( X — y ) is a retention potential that ensures that the propagated wavefunction x stays close to the optimized wavefunction y and fi is a mass parameter, k = co is the force constant of the retention potential. 

We can now begin to appreciate why the MC-SCF has been formulated using the X and A variables. Since one is using a Cl optimized wavefunction 3E/3A=0 and the second term in eq 36 vanishes. The term (3E/3T )( 3T/3qj) is the response of molecular orbitals to nuclear distortion. It has two parts one that arises from orbital reoptimization X 3E/3X and one that arises from orbital re-orthogonalization Y 3E/3Y in Box 1... 

Introducing the following notation for the electronic gradient and the electronic Hessian of the optimized wavefunction... 

For the optimized wavefunction we may thus identify the Lagrange multipliers in equation (34) with the Fock matrix in the MO basis. 

We have here first used the fact that the total derivatives of the SCF energy and of the SCF Lagrangian are identical for the optimized wavefunction. Next, we have invoked the Hellmann-Feynman theorem for the fully variational Lagrangian, thereby reexpressing the SCF molecular gradient as the partial derivatives of the Lagrangian. Finally, we have inserted the expression for the Lagrangian (equation 34). 

In spite of significant progress, some shortcomings that remain need to be noted. Quite often to obtain a stable wavefunction in Kohn-Sham DFT computations on systems containing transition metals the symmetry of the system wavefunction should be unconstrained [89]. Several cycles of consecutive wavefunction optimizations may be necessary to obtain a stable wavefunction. An optimized wavefunction should be checked for stability and, if unstable, re-optimized (several times) to obtain the lowest energy solution [89]. 

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