Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

and hence proves Eq. (C.36).

The above discussion is now generalized to arbitrary spin values. First, we note that twice application of the time-reversal operator leads the system back to

^2 ¼ ^ ^^^ 1 ^^^ 1 which proves that T 1. Now, by substituting Eq. (C.8) in TrT , TpT ,

^^^ 1 ^

and TST , we may show that U satisfies equations similar to Eqs. (C.24) and (C.25).

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs DðSÞðaÞ of the 3D rotational group SOð3Þ, where a is the rotational vector,

where ^gðaÞ is an operator of SOð3Þ. Let us then take ^gðaÞ to be a rotation by p around the y axis. Thus, from Eqs. (C.46) and (C.47), one gets

Comparing Eqs. (C.48)–(C.50) with Eq. (C.24), one obtains

where 2pj indicates a 2p rotation about the y axis. Thus, we have

and finally, by comparing with Eq. (C.51), one gets

Finally, for a system of n identical particles, the result is

APPENDIX D: DEGENERATE AND NEAR-DEGENERATE

VIBRATIONAL LEVELS

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3h configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The normal coordinates ðQx; Qy; QzÞ, the plane polar coordinates ðr; j; zÞ, and the Cartesian displacement coordinates ðxi; yi; ziÞ of the three nuclei ði ¼ 1; 2; 3Þ are related by [20,94]

where the coordinates ðQx; QyÞ are the doubly degenerate modes belonging to the E0 IRREP in D3h, and Qz belongs to the A01 one. Note that Qx is symmetric with respect to the xz plane, while Qy is antisymmetric.

The coordinates of interest to us in the following discussion are Qx and Qy, which describe the distortion of the molecular triangle from D3h symmetry. In the harmonic-oscillator approximation, the factor in the vibrational wave

function due to the two degenerate modes is then (except for a normalization constant and dependence on Qz) given by

where Hv2a and Hv2b are Hermite polynomials of order v2a and v2b, respectively; v2a and v2b are the vibrational quantum numbers, and a2 ¼ 2pn2=h, with n2 being the frequency of the degenerate mode.

Let us then consider the case where the degenerate mode is doubly excited. In this case, v2 ¼ v2a þ v2b ¼ 2 and the corresponding vibrational energy level will be triply degenerate with the associated wave functions being given by

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)–(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,

By using the plane polar coordinates defined in Eq. (D.1), one obtains

These new wave functions are eigenfunctions of the z component of the angular

momentum p^z ¼ ih qjq with eigenvalues mv2 ¼ þ2; 0; 2 in units of h. Thus, Eqs. (D.11)–(D.13) represent states in which the vibrational angular momentum

of the nuclei about the molecular axis has a definite value. When treating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in

The functions Pjlmj are associated Legendre polynomials of order jmj and degree l, and Lltþ1=2ðr2Þ are associated Laguerre polynomials of degree ðv 1Þ=2 in r2. In turn, the normalization factors are found to be

In the configuration space spanned by ðq1; q2; q3Þ, we may then define the vibrational angular momentum p through its classical components, that is,

p1 ¼ q2p3 q3p2 and its ð123Þ cyclic permutations ðD:38Þ

where pi are the conjugate momenta associated to qi ði ¼ 1; 2; 3Þ. The operators associated with p2 ¼ p21 þ p22 þ p23 and its projection pz (denoted M3 in [95])

the electronic energy of the system is in degenerate-state perturbation theory obtained by solving the secular equation

and the integrations are defined with respect to all the electronic coordinates.

^

Then, by developing He in a power series expansion of the normal coordinates, one gets

where we have considered only the dependence on the degenerate complex normal coordinates Qþ and Q . Substitution of Eq. (E.8) in Eqs. (E.6) and (E.7) gives

^þ ^ ^þ ^

must be totally symmetric, it follows that h1 , h1 , h2 , and h2 are multiplied by o, o , o2, and o 2, respectively. The integrals in Eqs. (E.9) and (E.10) will then be different from zero only if the integrands are invariant under all symmetry

^

operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Qþ and Q vanish in Hþþ and H . In turn,

the first term in Hþ and Hþ vanishes while one of the linear terms (Qþ for Hþ, and Q for Hþ) does not vanish. Thus, neglecting quadratic (and higher order) terms, one obtains

Clearly, Eq. (E.12) shows that to a first approximation the electronic energy varies linearly with displacements in r, increasing for one component state while decreasing for the other. Thus, the potential minimum cannot be at r ¼ 0. This is the statement of the Jahn–Teller theorem for a X3 molecule belonging to the D3h point group.

II. INVARIANT OPERATORS

We follow Thompson and Mead [13] to discuss the behavior of the electronic Hamiltonian, potential energy, and derivative coupling between adiabatic states

in the vicinity of the D3h conical intersection. Let ^ be an operator that

A

^

transforms only the nuclear coordinates, and A be one that acts on the electronic degrees of freedom alone. Clearly, the electronic Hamiltonian satisfies

since, if c is an eigenfunction of He, Eq. (E.13) just expresses the fact that Ac is

an eigenfunction of the transformed Hamiltonian with the same eigenvalue (for

^

If Eq. (E.14) is satisfied for all elements of some point group G, A will be an invariant operator [13] (the Hermitian conjugate as well as the sum and/or product of two invariant operators are also invariant operators). Such an operator can be expanded in the form

where Q gs is a nuclear coordinate transforming as the g th component of the IRREP of G, the index s refers to different occurrences of the same IRREP, and