Energy minimization, Hartree-Fock equations

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

In the literature we may find the procedure for creating localized Hartree-Fock orbitals via an energy minimization based on a Cl procedure employing monoexcitations (see for instance Reference [24]). The scheme starts from a set of given (guess) orbitals and solves iteratively the Hartree-Fock equations via the steps ... 

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 ... 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

The next step in deriving the Hartree-Fock equations is to express the energy of the molecule or atom in terms of the total wavefunction T the energy will then be minimized with respect to each of the component molecular (or atomic an atom is a... 

Using this wavefunction the total energy of the electron system is minimized with respect to the choice of spin-orbitals under the constraint that the spin orbitals are orthogonal. The variational procedure is applied to this minimization problem and the result is the so called Hartree-Fock equations ... 

The Hartree-Fock equation is obtained by requiring that the orbitals minimize the expectation value of the energy. Those orbitals satisfy... 

By minimizing the energy of d>, in Eq. (3.12), we obtain a set of coupled integro-differential equations, the Hartree-Fock equations, which may be expressed in the following form for closed-shell systems (for open-shell cases see Szabo and Ostlund, 1989) ... 

The only variational degree of freedom concerns the orbital set, which is therefore chosen to minimize the energy expectation value with the constraint that the orbitals remain orthonormal. This leads to a set of Euler equations that in turn lead to the Hartree-Fock equations, finally giving the (f)/ set although in an iterative way because the Hartree-Fock equations depend on the orbitals themselves. This dependency arises from the fact that the HF equations are effective one-electron eigenvalue equations... 

The best possible variationally determined wavefunction of this form is that in which both the spacial orbitals total wavefunction be normalized. For such a wave-function, the variational method leads to orbital equations similar to the Hartree-Fock equations (10). These are.-26... 

The Hartree-Fock method involves the optimization - via energy minimization -of a single Slater determinant wavefunction. This process is usually carried out by solving the single-particle Hartree-Fock equations. From the Hartree-Fock orbitals one can construct the Hartree-Fock density pur-... 

Now that the description of the electronic Hamiltonian, Eq. [4], has been determined and the wavefunction, Eq. [6], has been defined, the effective electronic energy can be found by use of the variational method. In the variational method the best wavefunction is found by minimizing the effective electronic energy with respect to parameters in the wavefunctions. Using this idea, Fock and Slater simultaneously and independently developed what is now known as the Hartree—Fock equations. Note that we now make explicit reference only to the spatial orbitals < >. The only time we make reference to spin is that we will fill the orbitals according to the aufbau principle and place two electrons in each spatial orbital. 

Other examples of optimizing functions that depend quadraticaUy of the parameters include the energy of Hartree-Fock (HF) and configuration interaction (Cl) wave functions. Minimization of the energy with respect to the MO or Cl coefficients leads to a set of linear equations. In the HF case, the Xy coefficients depend on the parameters Ui, and must therefore be solved iteratively. In the Cl case, the number of parameters is typically 10 -10 and a direct solution of the linear equations is therefore prohibitive, and special iterative methods are used instead. The use of iterative techniques for solving the Cl equations is not due to the mathematical nature of the problem, but due to computational efficiency considerations. 

Section 3.2 constitutes a derivation of the results of the previous section. The order of presentation of these two sections is such that the derivations of Section 3.2 can be skipped if necessary. For a fuller appreciation of Hartree-Fock theory, however, it is recommended that the derivations be followed. We first present the elements of functional variation and then use this technique to minimize the energy of a single Slater determinant. A unitary transformation of the spin orbitals then leads to the canonical Hartree-Fock equations. 

Just as one can show that the orthononnal orbitals that minimize the Hartree-Fock expression for the molecular energy satisfy the Fock equation (14.25), one can show that the Kohn-Sham orbitals that minimize the expression (16.46) for the molecular ground-state energy satisfy (for a proof, see Parr and Yang, Section 7.2) ... 

The matrix elements in Eqs. 2.64 through 2.66 are referred to as the core integral, the Coulomb integral, and the exchange integral, respectively. Minimization of the trial energy by varying the 0( under the constraint that the basis functions remain orthonormal leads to the Hartree-Fock equation [6]... 

The derivation of the Hartree-Fock equations in the case of polarizable classical sites is slightly more laborious. The reason is that the MM contribution to the polarization energy depends on the QM charge distribution through the induced moments. In order to get the correct Hartree-Fock equations, it is necessary to include in the energy minimization procedure. 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

The only term for which no explicit form can be given, i. e the big unknown, is of course Exc- Similarly to what we have done within the Hartree-Fock approximation, we now apply the variational principle and ask what condition must the orbitals cp fulfill in order to minimize this energy expression under the usual constraint of ((p I (pj) = ,j The resulting equations are (for a detailed derivation see Parr and Yang, 1989) ... 

Thus four of the seven lowest H20 MOs are linear combinations of the four a, symmetry orbitals listed above, and are a, MOs similarly, the two lowest b2 MOs are linear combinations of 02p and H,1j — H21.s, and the lowest bx MO is (in this minimal-basis calculation) identical with 02px. The coefficients in the linear combinations and the orbital energies are found by iterative solution of the Hartree-Fock-Roothaan equations. One finds the ground-state electronic configuration of H20 to be... 

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