Energy calculation from wavefunction

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

The process is continued for k cycles till we have a wavefunction and/or an energy calculated from that are essentially the same (according to some reasonable criterion) as the wavefunction and/or energy from the previous cycle. This happens when the functions i/ (l), i//(2),. .., j/(n) are changing so little from one cycle to the next that the smeared-out electrostatic field used for the electron-electron potential has (essentially) ceased to change. At this stage the field of cycle k is essentially the same as that of cycle k — 1, i.e. it is consistent with this previous field, and so the Hartree procedure is called the self-consistent-field-procedure, which is usually abbreviated as the SCF procedure. 

However, it would be a serious error to confuse electronegativity or ionization energy with hardness . The clearest counter-example is Tl(III) which is a rather soft central atom but which must have a rather high ionization energy. There are other physical properties which accentuate the opposite inequality signs of (1) even more, for instance the electric polarizability. Table 1 contains many values for this quantity a in the unit 10-24 cm3. Gaseous atoms and positive ions have polarizabilities which can be calculated from wavefunctions (9) by evaluating the sum of matrix elements of induced dipole moment ... 

Combination of the Is and 2s atomic orbitals with a or spin functions yields four spin orbitals for construction of a Slater determinant we need only three. The energy calculated from the wavefunction is the same whether we use the spin orbital 2s (r)a (s) or ground state of the Li atom (or any other Group 1 metal), like the ground state of the H atom, is doubly degenerate. 

When discussing the He atom, we stressed that a wavefunction written as the product of two atomic orbitals is inherently wrong since it fails to reflect the fact that repulsion between the electrons tends to keep them far apart. The energy calculated from such a wavefunction will therefore be higher than that of the real atom. What was true for the two-electron atom, is equally true for a two-electron molecule. Like atomic orbital calculations, molecular orbital calculations on moiecules containing two or more electrons are inherently wrong. 

Variation theory states that the energy calculated from any trial wavefunction will never be less than the true ground-state energy of the system. This means that the smaller the value of e, the closer it is to the tme ground-state energy of the system and the more xp sai represents the true ground-state wavefunction. The trial wavefunction is set up with one or more adjustable parameters, pi, making the function flexible to minimize the value of e. An n number of adjustable parameters will set up an n number of differential equations ... 

Figure 13. Left ligand field energy-level diagram calculated for plastocyanin. Center contains energies and wavefunctions of the copper site. Energy levels determined after removing the rhombic distortions to give and C symmetries are shown in the left and right columns, respectively (from Ref. 11). Right electronic structural representation of the plastocyanin active site derived from ligand field calculations (from Ref. 11).

We have m x m equations because each of the m spatial MO s i// we used (recall that there is one HF equation for each ip, Eqs. 5.47) is expanded with m basis functions. The Roothaan-Hall equations connect the basis functions (p (contained in the integrals F and S, Eqs. 5.55, above), the coefficients c, and the MO energy levels . Given a basis set energy levels e. The overall electron distribution in the molecule can be calculated from the total wavefunction P, which... 

The key process in the HF ab initio calculation of energies and wavefunctions is calculation of the Fock matrix, i.e. of the matrix elements Frs (Section 5.2.3.6.2). Equation (5.63) expresses these in terms of the basis functions and the operators //core, J and K, but the J and K operators (Eqs. 5.28 and 5.31) are themselves functions of the MO s i// and therefore of the c s and the basis functions Fock matrix to be efficiently calculated from the coefficients and the basis functions without explicitly evaluating the operators J and K after each iteration. This formulation of the Fock matrix will now be explained. 

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