The Electronic Problem

Our main interest in this book is finding approximate solutions of the non-relativistic time-independent Schrodinger equation 

In the above equation, is the ratio of the mass of nucleus A to the mass of an electron, and is the atomic number of nucleus A. The Lapladan operators Vf and involve differentiation with respect to the coordinates of the ith electron and the Aih nucleus. The first term in q. (2.2) is the operator for the kinetic energy of the electrons the second term is the operator for the kinetic energy of the nuclei the third term represents the coulomb attraction between electrons and nuclei the fourth and fifth terms represent the repulsion between electrons and between nuclei, respectively. 

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From this point on, we will focus entirely on the electronic problem. We will omit the superscripts on all operators and functions. 

But we can carry forward the knowledge of the Bom-Oppenheimer approximation gained from Chapter 2 and focus attention on the electronic problem. Thus... 

If we want to calculate the potential energy curve, then we have to change the intemuclear separation and rework the electronic problem at the new A-B distance, as in the H2 calculation. Once again, should we be so interested, the nuclear problem can be studied by solving the appropriate nuclear Schrodinger equation. This is a full quantum-mechanical equation, not to be confused with the MM treatment. 

The x s can individually be Is, 2s, 2p,. .. atomic orbitals. The lowest-energy solution will be when the x s correspond to Is orbitals on each of the two hydrogen atoms, the next-highest-energy solution will be when one of the ( s is a Is, the other a 2s atomic orbital, and so on. Possible solutions of the electronic problem, with the two H atoms at infinity, are shown in Table 4.1. 

The attractive potential exerted on the electrons due to the nuclei - the expectation value of the second operator VNe in equation (1-4) - is also often termed the external potential, Vext, in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields etc. From now on we will only consider the electronic problem of equations (1 -4) - (1 -6) and the subscript elec will be dropped. 

The QM/MM methodology [1-7] has seen increasing application [8-16] and has been recently reviewed [17-19], The classical solvent molecules may also be assigned classical polarizability tensors, although this enhancement appears to have been used to date only for simulations in which the solute is also represented classically [20-30], The treatment of the electronic problem, whether quantal, classical, or hybrid, eventually leads to a potential energy surface governing the nuclear coordinates. 

Here G(vj, v2, v3) is the level energy in wave number units (as far as possible we follow the notation of Herzberg, 1950) and the constants in Equation (0.1) are given in Table 0.1. As usual the vs are the vibrational quantum numbers of S02 and rather high (above 10) values can be reached using the SEP technique. Equation (0.1) provides a fit to the observed levels to within an error below 10 cm 1, which is almost the experimental accuracy. We need, however, to be able to relate the parameters in this expansion directly to a Hamiltonian. The familiar way of doing this proceeds in two steps. First, the electronic problem is solved in the Bom-Oppenheimer approximation, leading to the potential for the... 

We can make further approximations to simplify the NRF of the Hamiltonian presented in equation (75) for non-dynamical properties. For such properties, we can freeze the nuclear movements and study only the electronic problem. This is commonly known as the clamped nuclei approximation, and it usually is quite good because of the fact that the nuclei of a molecule are about 1836 times more massive than the electrons, so we can usually think of the nuclei moving slowly in the average field of the electrons, which are able to adapt almost instantaneously to the nuclear motion. Invocation of the clamped nuclei approximation to equation (75) causes all the nuclear contributions which involve the nuclear momentum operator to vanish and the others to become constants (nuclear repulsion, etc.). These constant terms will only shift the total energy of the system. The remaining terms in the Hamiltonian are electronic terms and nuclear-electronic interaction contributions which do not involve the nuclear momentum operator. 

In terms of these conditions, a fc-particle hierarchy of approximations can be defined, with Hartree-Fock as the one-particle approximation for closed-shell states. Unfortunately, the stationarity conditions do not determine the fully, and for their constmction additional information is required, which essentially guarantees -representability. Nevertheless, the fe-particle hierarchy based on the irreducible stationarity conditions opens a promising way for the solution of the -electron problem. 

An alternate route to the electronic problem starts off at the HL limit and adds in the effect of the overlap interactions between the atomic orbitals (f and ( >2(= /9). Our treatment is adapted from that in reference 8. The complete set of singlet states of the problem are... 

The index k(n) recalls that the nuclear fluctuation quantum states in eq.(l 1) are determined by the electronic quantum state via potential energy Een(7 )- Once the electronic problem is fully solved, via a complete set ofeq.(5), it is not difficult to see that pTif nk) multiplied by the box-normalized wave solutions (see p. 428, ref. [17] 2nd ed.) are eigenfunctions ofthehamiltonian H0and, for stationary global momentum solutions, the molecular hamiltonian is also diagonalized thereby solving eq. (2). 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

As we have already mentioned in Section 2.2, a huge number of plane waves is required for all electron calculations, in order to properly describe the core region. Therefore, different approaches are used to exclude the core states from the electronic problem to be solved. There is no universal rule concerning the cutoff energy for each method different values are typically required for convergence. 

Research Education Association, The Electronics Problem Solver (Research and Education Association, Piscataway, NJ, 2000). 

We use the operator on the left-hand side of equation (7.169) as the zeroth-order vibrational Hamiltonian. The remaining terms in the effective electronic Hamiltonian, given for example in equations (7.124) and (7.137), are treated as perturbations. In a similar vein to the electronic problem, we consider only first- and second-order corrections as given in equations (7.68) and (7.69) to produce an effective Hamiltonian 3Q, which is confined to act within a single vibronic state rj, v) only. Once again, the condition for the validity of this approximation is that the perturbation matrix elements should be small compared with the vibrational intervals. It will therefore tend to fail for loosely bound states with low vibrational frequencies. 

Equation (25) follows after differentiation and application of Stirling s formula (n = N/Nm, nd = Nd/Nm, Nm = Avogadro s number /f = NmAg°d). It is worthy to note that the strict result (see l.h.s. of Eq. 25) which is formally valid also for higher concentrations, is of the Fermi-Dirac type. This is due to the fact that double occupancy is forbidden and hence the sites are exhaustible similar as it is the case for the quantum states in the electronic problems. 

The Ge-Co(CO)4 is equivalent to a Ge-R three-orbital-two-electron fragment hence the cluster has eight sep/68 eve instead of the expected seven sep/66 eve. Hence, the electronic problem is the one dealt with in Section 5.2.1 where octahedral Fe4(CO)i2(+4-PR)2 was considered. 

The coherent potential approximation (CPA) was originally developed by Soven for the electronic problem and by Taylor for the phonon problem. The basic assumption of CPA is that a random medium with diagonal disorder can be described by an efiiective periodic Hamiltonian of the type... 

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