Minimality of the Hamiltonian

we will show that the dot product A (y — y) in Inequality (5.16) is constant so that the inequality also holds at ti, the time of pulse perturbation. This outcome will reveal that at any time when the optimal control is continuous, the corresponding Hamiltonian is minimum, and any control perturbation does not decrease the Hamiltonian further. Finally, we will extend this minimality to times when the optimal control is not continuous. 

the dot product A (y — y) is constant so that Inequality (5.16) holds throughout the time interval In particular. 

Note that ti is any arbitrary time instant when the optimal control is continuous. Hence the optimal control, whenever continuous, minimizes the Hamiltonian. 

Belonging to the class defined in Section 5.2.1 (p. 126), the optimal control u could be discontinuous in the interval [0, tf]. Therefore, we need to show that even when u is discontinuous, it minimizes the Hamiltonian. We will derive this result in the following three steps  

We will show that the function h t) is continuous throughout the interval 

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It is easily seen that for such trial functions the minimization of the Hamiltonian K, Eq. (5.1), may be replaced by the minimization of a specified nonlinear functional 6(0) of the molecular states 0 alone. In the following we refer to either formulation as seems convenient. This argument also enables one to connect these field theoretical models with the earlier suggestion of mine that molecular structure states can be associated with those solutions of the Schrodinger equation for the full molecular Hamiltonian ft that satisfy certain subsidiary conditions3,35), if the latter are associated with the nonlinearity in the functional 6(0). As we shall see, it may happen that 6(0) has two degenerate minima and it is in this sense that the dynamics gives rise to a double-well structure. 

In this chapter, we will present the proof of the minimum principle. The minimum principle uses a positive multiplier for the objective functional in the Hamiltonian formulation. With this provision, the minimum principle concludes that the minimum of the problem requires minimization of the Hamiltonian in an optimal control problem whose minimmn needs to be determined. 

We can now write down the Hartree-Fock one-electron hamiltonian in the D— oo limit. As usual, we will write the hamiltonian as one for the probability amplitude, and will remove the dominant dimension-dependence of the solutions through use of appropriately scaled units. As discussed in the previous section, this means that energies will be in units of 4/(D —1) haxtrees, and distances in units of D(D—l)/6 Bohr radii. The symmetry assumption allows us to equate all electron-nucleus distances, and to constrain the electrons to positions directly above the nuclei, prior to minimization of the hamiltonian. Also, the Hartree-Fock approximation allows dihedral angles to be fixed at 90°. With these scalings and simplifications, the D— oo limit hamiltonian can be written... 

For the calculation of the LDA ground-state one can proceed either via the direct" methods, i.e. via the glocal minimization of the total free energy with respect to the electronic degrees of freedom, or via the the diagonalization (for large PW basis-sets necessarily iterative diagonalization) of the KS Hamiltonian in combination with an iterative update of chai ge-density and potential. 

Given the trial wavefunction - the Slater determinant eq. (11.37) - we then use the variational principle to minimize the energy - the expectation value of the Hamiltonian H - with respect to the orbital coefficients cy (eq. (11.39)). This leads after a fair amount of algebra to the self-consistent Hartree-Fock equations ... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

The corresponding formulation was made by von Neumann2 for quantum mechanics. This entropy-maximizing (or information-minimizing) principle is the most direct path to the canonical distribution and thus to the whole equilibrium theory. It is understood that the extremalizing is conditional, i.e., certain expected values, such as that of the Hamiltonian, are fixed. 

The trajectory q pl (t) is determined by minimizing S in (20) on the set of all classical deterministic trajectories determined by the Hamiltonian H (37), that start on a stable limit cycle as t — —oo and terminate on a saddle cycle as t > oo. That is, qopt(t) is a heteroclinic trajectory of the system (37) with minimum action, where the minimum is understood in the sense indicated, and the escape probability assumes the form P exp( S/D). We note that the existence of optimal escape trajectories and the validity of the Hamiltonian formalism have been confirmed experimentally for a number of nonchaotic systems (see Refs. 62, 95, 112, 132, and 172 and references cited therein). 

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

Because (4> ff S) is not in itself a variational expression, its unconstrained minimum value is not simply related to an eigenstate of the Hamiltonian Hv defined by v in Eq.(3), whereas Eq.(2) defines F[p only for such eigenstates. Any arbitrary trial function J —> can be expressed in the form + Aca with ca = 1. If the minimizing trial function in Eq.(3) were not an eigenfunction of Hv, then for some subset of trial functions, using the Brueckner-Brenig condition,... 

The equations of the HF approximation are derived by taking expectation values of the Hamiltonian with respect to a determinantal wave function written in terms single-particle states, incorporating the orthonormality condition by means of a Lagrange multiplier, and minimizing the expression... 

By its very nature, minimization of En is more expensive than the maximization of S, because it requires the construction of quantities corresponding to applications of the hamiltonian operator, but this may be achieved by adapting the efficient procedures already available in various CASSCF codes. It turns out, however, that the two sets of orbital representations tend to be rather similar, and so maximization of tends to be preferred. In either case, the actual optimization uses reliable Newton-Raphson-like procedures that utilize first and second derivatives. 

The coefficients Cpt are obtained by minimizing the expectation value of the Hamiltonian for the wavefunction V from equation (2). In the more commonly occurring closed-shell case this gives the secular equation... 

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