The Hamiltonian Operator

The three-dimensional Schrodinger wave equation for a particle of mass m moving in a potential energy field V was written in Chapter 5 as  

The wavefunction, y/, and the potential energy, V, are both functions of the coordinates of the particle, x, v and z. The total energy of the particle is represented by E, which must be constant. 

The expression in parentheses is known as the Hamiltonian operator and is given the symbol H. It contains instructions for the mathematical manipulation of whatever function follows it (for example, differentiate twice with respect to x, y and z). The cap over the H is a reminder that this term is an operator, and not simply a multiplier. With this shorthand notation the Schrodinger equation can be written simply as  

The Hamiltonian operator is named after Sir William Rowan Hamilton, an Irish mathematician, who devised an alternative form of Newton s equations of motion involving a function H, known as the Hamilton function. For most classical systems, H turns out to be simply the total energy of the system expressed in terms of the coordinates of the particles and their conjugate momenta. The kinetic energy of a single particle of mass m can be written as  

The entirely classical Hamiltonian function, H, can be converted into the Hamiltonian operator, H, by applying some simple rules, which can be stated as follows  

So what function does H serve, and why is it impossible to solve Eq. 3.1 when 1 The Hamiltonian is the total energy operator and describes all the kinetic and potential energy terms within a molecule, namely  

With a little thought, it becomes apparent how we can simplify matters. There is a massive difference in the masses and, hence, velocities of electrons and nuclei, to the extent that we can treat their motions separately. This is the Bom-Oppenheimer approximation. From the viewpoint of the electrons the nuclei appear to be stationary (i.e. term (ii) equates to zero), and if the nuclear positions are fixed, then the 

From a computational perspective, the ideal would be to solve Schrodinger s equation one electron at a time, to give N one-electron functions. These equations would then be summed to generate the complete multi-electron solution. But the electron-electron repulsion (term (iv)) presents us with a serious problem it states that the behavior of each electron in the system influences that of all the others. It is this correlated behavior that means that we cannot describe each electron individually without making some more approximations, and it is for this reason that we cannot obtain exact solutions to the Schrodinger equation for multi-electron systems. 

But there is more than just electron correlation to worry about. We know that each orbital can accommodate a maximum of two electrons, provided they are of opposite spin (denoted a (spin up, ) and (3 (spin down, )). This requirement to pair up electrons of opposite spins, while keeping electrons of parallel spins apart, is known as electron exchange, and must also be included in any accurate molecular description. This is achieved by Fock theory, which makes use of an antisymmetric wavefunction. In this context, antisymmetric means that the wavefunction changes its sign when the coordinates of two electrons are exchanged. We can demonstrate how this is done by considering two electrons, labeled a and b, both with spins up. An antisymmetric wavefunction, for this system would be  

Note how — ab is obtained upon exchanging r and r2- If we now try and put the two electrons in the same place, i.e. set r = r2, then we see that the wavefunction collapses to zero. Thus our wavefunction obeys Pauli s Exclusion Principle. 

We are interested in what happens when a magnetic moment fJt interacts with an applied magnetic field B0—an interaction commonly called the Zeeman interaction. Classically, the energy of this system varies, as illustrated in Fig. 2.1a, with the cosine of the angle between l and B0, with the lowest energy when they are aligned. In quantum theory, the Zeeman appears in the Hamiltonian operator 

FIGURE 2.1 (a) Relative orientations of magnetic moment and magnetic field B0. (b) Quan- 

Here I is interpreted as a quantum mechanical operator. From the general properties of spin angular momentum in quantum mechanics, it is known that the solution of this Hamiltonian gives energy levels in which 

There are thus 21+1 energy levels, each of which may be thought of as arising from an orientation of p. with respect to B0 such that its projection on B0 is quantized (See Fig. 2.1 b). Equation 2.5 shows that the energy separation between the states is linearly dependent on the magnetic field strength. 

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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... 

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... 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

One can show that the expectation value of the Hamiltonian operator for the wavepacket in equation (A3.11.71 is ... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

The sum of two operators is an operator. Thus the Hamiltonian operator for the hydrogen atom has — j as the kinetic energy part owing to its single election plus — 1/r as the electiostatic potential energy part, because the charge on the nucleus is Z = 1, the force is atrtactive, and there is one election at a distance r from the nucleus... 

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation... 

The first of these equations is ealled the time-independent Sehrodinger equation it is a so-ealled eigenvalue equation in whieh one is asked to find funetions that yield a eonstant multiple of themselves when aeted on by the Hamiltonian operator. Sueh funetions are ealled eigenflinetions of H and the eorresponding eonstants are ealled eigenvalues of H. 

Once a wave function has been determined, any property of the individual molecule can be determined. This is done by taking the expectation value of the operator for that property, denoted with angled brackets < >. For example, the energy is the expectation value of the Hamiltonian operator given by... 

Schrddinger s equations are usually written in a more succinct manner by invoking the Hamiltonian operator H, so for example the time-dependent equation for a single particle... 

The Hamiltonian operator is the quantum-mechanical analogue of the energy, and we say that the allowed values of the energy, the , above, are the eigenvalues... 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities. The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rp, and the scalar distance between the electron and nucleus B is tb- I will write / ab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb-It is an easy step to write down the Hamiltonian operator for the problem... 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is... 

The Hamiltonian operator for such a system may be written under the form... 

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 ... 

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