The Hamiltonian

One of the most confusing aspects of quantum mechanics can be the notation. We will do our best here to keep it clear and simple. From this point forward in the chapter, wave-functions (the eigenfunctions) y/, and 0 will be defined as follows. The symbol F will rep- 

CHAPTER 14 ADVANCED CONCEPTS IN ELECTRONIC STRUCTURE THEORY 

UPPER CASE = Nuclei, lower case = electrons m or /W = mass  

O = Kinetic energy of nuclei 0 = Nuclear-nuclear repulsions = Kinetic energy of electrons O = Nuclear-electron attraction 0 = Electron-electron repulsion 

The general form of the Hamiltonian for any molecule, along with the meaning of each term. 

As we know the quantum mechanical Hamiltonian operator determines the properties of a spin system as follows, 

In a solid, if the electrons outside of closed shells are not s electrons, the Hamiltonian for localized outer electrons with the same quantum numbers n, l is 

Solution of the Schroedinger equation, H p = Ef, appropriate to this problem has only been accomplished by means of successive perturbation calculations. The zero-order approximation is a spherical approximation in which a given outer electron is assumed to move in the average potential of the other outer electrons as well as of the core electrons. Then the free-ion Hamiltonian becomes 

Investigation of the experimental data indicates that the strengths of the ligand fields fall into three groups  

Strong fields with ALS Aei Af/, where the electrostatic splitting is Aei 104 cm-1 This situation occurs only with d electrons, 

Medium fields with ALs Ac/ Aei It is the relative energies that are important the multiplet splitting may vary from 10 l to 104 cm-1 (see Sommerfeld s (594) fine structure formula). In this case Fei is the first perturbation effect, and the cubic part of Vcf is treated as a perturbation before spin-orbit effects are calculated. For distortions from cubic symmetry, it is necessary to consider carefully the relative magnitudes of Vt (that part of VCf due to departures from cubic symmetry) and Vis S. Weak fields with ACf ls - In this case the ligand fields merely perturb the multiplet structure of the free atom. 

I appealed to your good sense when I asked you to believe that the Hamiltonian for a particle in an external electric field could be written 

Derivation of this equation is actually far from straightforward. In electromagnetism, we describe static fields by the electric field E and the magnetic induction B. For our purposes, we need to enquire about the potentials rather than the fields, and these are defined by 

We have met the electrostatic potential 4 in earlier chapters. The vector potential A is a fundamental construct in electromagnetism (HinchUffe and Munn, 1985). 

Imagine a free particle with charge Q and mass m. The Hamiltonian is 

The quantity p — QA is called a generalised momentum. It appears in both classical electromagnetism and quantum mechanics. In the Schrbdinger picture, we make the substitution 

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Eigen function In wave mechanics, the Schrodinger equation may be written using the Hamiltonian operator H as... 

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

The time-dependent Sclirodinger equation allows the precise detemiination of the wavefimctioii at any time t from knowledge of the wavefimctioii at some initial time, provided that the forces acting witiiin the system are known (these are required to construct the Hamiltonian). While this suggests that quaiitum mechanics has a detemihiistic component, it must be emphasized that it is not the observable system properties that evolve in a precisely specified way, but rather the probabilities associated with values that might be found for them in a measurement. 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

At this point, it is appropriate to make some conmrents on the construction of approximate wavefiinctions for the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is given by the general fonn... 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

Although it may be impossible to solve the Sclirodinger equation for a specific choice of the Hamiltonian, it is... 

In perturbation theory, the Hamiltonian is divided into two parts. One of these eorresponds to a Selirodinger equation that ean be solved exaetly... 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

The characteristic of the Darlhig-Deimison couplmg is that it exchanges two quanta between the synmietric and antisynmietric stretches. This means that the individual quantum mimbers n are no longer good quaiitum mimbers of the Hamiltonian containing However, the total iiumber of stretch quanta... 

Consider the polyad = 6 of the Hamiltonian ( Al.2.7). This polyad contains the set of levels conventionally assigned as [6, 0, ], [5, 1],. . ., [0, 6], If a Hamiltonian such as ( Al.2.7) described the spectrum, the polyad would have a pattern of levels with monotonically varymg spacing, like that shown in figure Al.2.8. 

One of the most significant achievements of the twentieth century is the description of the quantum mechanical laws that govern the properties of matter. It is relatively easy to write down the Hamiltonian for interacting fennions. Obtaining a solution to the problem that is sufficient to make predictions is another matter. 

Using the Hamiltonian in equation Al.3.1. the quantum mechanical equation known as the Scln-ddinger equation for the electronic structure of the system can be written as... 

One can utilize some very simple cases to illustrate this approach. Suppose one considers a solution for non-interacting electrons i.e. in equation A1.3.1 the last temi in the Hamiltonian is ignored. In diis limit, it is... 

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

The Hamiltonian matrix factorizes into blocks for basis functions having connnon values of F and rrip. This reduces the numerical work involved in diagonalizing the matrix. 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

For the Hamiltonian //we identify a synnnetry group, and this is a group of synnnetry operations of /7a synnnetry operation being defmed as an operation that leaves //invariant (i.e., that coimmites with //). In our example, the synnnetry group is K (spatial). 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

The Hamiltonian matrix will be block diagonal in this basis set. There will be one block for each irreducible representation of the synnnetry group. 

The possible types of symmetry for the Hamiltonian of an isolated molecnle in field-free space (all of them are discussed in more detail later on in the article) can be listed as follows ... 

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