Other Electronic Operators

By extension of Exercise 6-1, the Hamiltonian for a many-electron molecule has a sum of kinetic energy operators — V, one for each electron. Also, each electron moves in the potential field of the nuclei and all other electrons, each contiibuting a potential energy V,... 

Guess the average potential that a specific electron would feel coming from the other electrons that is, you guess at the Fock operator. 

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The one electron operator h, describes the motion of electron i in the field of all the nuclei, and gy is a two electron operator giving the electron-electron repulsion. We note that the zero point of the energy corresponds to the particles being at rest (Tc = 0) and infinitely removed from each other (Vne = Vee = V n = 0). 

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron, the attraction to all the nuclei and the repulsion to all the other electrons (via the J and K operators). Note that the Fock operator is associated with the variation of the total energy, not the energy itself. The Hamilton operator (3.23) is not a sum of Fock operators. 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

The Schrodinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Bom-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

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