Three-body Algebraic Theory

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrödinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate r1,r2,... and momentum p1, p2, . . . , boson creation and annihilation operators, b†iα, biα. The index i runs over the number of relevant degrees of freedom, while the index α runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ≠ j, . . . [biα, b†jα´] = 0, [biα, bjα´] = 0,. . . . . .[bjα, b†iα´] = 0, [b†jα, b†iα´] = 0,. . . . . . [biα, b†iα´] = ẟαα´, [biα, b†iα´] = 0, [b†iα, b†iα´] = 0. . . .

Summary of Elements of Algebraic Theory

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.

Many-Body Algebraic Theory

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r1, r2, r3, . . . , quantized through the algebra . . .G ≡ U1(4) ⊗ U2(4) ⊗ U3(4). . . . . . .(6.1). . . becomes very cumbersome. Each time a U(4) algebra is added one must go through the recoupling procedure using Racah algebra, which, although feasible, is in practice very time consuming. An alternative treatment, which can be carried out for molecules with any number of atoms, is that of separating vibrations and rotations as already discussed in Sections 4.2-4.5 for triatomic molecules. For nonlinear molecules, there are three rotational degrees of freedom, described by the Euler angles α, β, γ of Figure 3.1, and thus there remain 3n − 6 independent vibrational degrees of freedom, where n is the number of atoms in the molecule. For linear molecules, there are two rotational degrees of freedom, described by the angles α, β, and thus there remain 3n − 5 independent vibrational degrees of freedom, some of which (the bending vibrations) are doubly degenerate. In this alternative treatment, the algebraic theory of polyatomic molecules consists in the separate quantization of rotations and vibrations. Each bond coordinate is then a scalar, and the corresponding algebra is that of U(2). In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book has the point group symmetry D6h. A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates.

Four-Body Algebraic Theory

In tetratomic molecules, there are three independent vector coordinates, rl, r2, and r3, which we can think of as three bonds. The general algebraic theory tells us that a quantization of these coordinates (and associated momenta) leads to the algebra . . .G = U1(4) ⊗ U2(4) ⊗ U3(4). . . . . . .(5.1). . . As in the previous case of two bonds, discussed in Chapter 4, we introduce boson operators for each bond . . .σ †1, π†1μ , μ = 0, ±1 ,. . . . . .σ †2, π†2μ , μ = 0, ±1 ,. . . . . .σ †3, π†3μ , μ = 0, ±1 ,. . . . . .(5.2). . . together with the corresponding annihilation operators σ1, π1μ, σ2, π2μ, σ3, π3μ. The elements of the algebras Ui(4) are the same as in Table 2.1, except that a bond index i = 1, 2, 3 is attached to them.

The Wave Mechanics of Diatomic Molecules

The spectroscopy of diatomic molecules (Herzberg, 1950) serves as a paradigm for the study of larger molecules. In our presentation of the algebraic approach we shall follow a similar route. An important aspect of that presentation is the discussion of the connection to the more familiar geometrical approach. In this chapter we survey those elements of quantum mechanics that will be essential in making the connection. At the same time we also discuss a number of central results from the spectroscopy of diatomic molecules. Topics that receive particular attention include angular momentum operators (with a discussion of spherical tensors and the first appearance of the Wigner-Eckart theorem which is discussed in Appendix B), transition intensities for rovibrational and Raman spectroscopies, the Dunham expansion for energy levels, and the Herman- Wallis expansion for intensities. Two-body quantum mechanical systems are conveniently discussed by transforming to the center-of-mass system. The momentum (differential) operator for the relative motion is . . .p = − iħ∇ . . . . . .(1.1). . . where ∇ is the gradient operator whose square ∇ • ∇ is the Laplacian and, as usual, i2 = − 1 and ħ is Planck’s constant/2π.

Prologue to the Future

At this point, we hope to have demonstrated that the algebraic approach provides a viable method for the quantitative description of molecular vibrotational spectra. Chapters 4 (triatomic molecules, both linear and bent) and 5 (linear tetratomic molecules) and Appendix C provide extensive documentation for the quantitative applications, while Chapter 6 shows that larger molecules can also be treated. Throughout, but most particularly in Chapter 7, we have sought to forge a link with the more familiar geometrical approach. It is precisely our requirement that even in zeroth order the Hamiltonian with which we start describes an anharmonic motion, which makes this link not trivial. The advantage of our approach in providing, even in zeroth order, high overtone spectra that are typically more accurate than 10 cm−1, should not be overlooked. Yet much remains to be done. In this chapter we look to the future: Where and why do we think that the algebraic approach will prove particularly advantageous? Of course, what we really hope for is to be surprised by unexpected new developments and applications. Here, however, is where we are certain that some of the future progress will be made, with special reference to the spectroscopy of higher-energy states of molecules. One area of spectroscopy where the Hamiltonian in matrix form is the route of choice is that of large polyatomics, particularly so when in an electronically excited state (see Note 3 of the Introduction). Such states are isoenergetic with very high vibrational overtones of the ground electronic states so that a fully geometrical approach is impractical. Even at lower energies, the exceedingly high density of vibrational states strongly favors an alternative approach, and the use of model Hamiltonian matrices is not uncommon. Such model matrices are introduced in order to account for the regularities that often survive in the observed spectrum. One such striking feature is often referred to as a “clump” (Hamilton, Kinsey, and Field, 1986). Consider a pure vibrational progression of states, as can be observed in stimulated emission spectroscopy.