Symmetry coordinates, Hamiltonian

The Hamiltonian operators can be written in terms of the symmetry coordinates... 

The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total 0(4) quantum numbers (ti, x2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. 

Figure 4 Convergence plots of perturbative energies for the 7 = 0 curvilinear normal coordinate Hamiltonian of H2CO. The energy difference E(, — E is plotted as a function of for the energy eigenstates with a, symmetry and N, 8. Note the difference in the scales of the y-axis. (From Ref. 49.)...

The occurrence of electron degeneracy brings two effects. First, the molecular Hamiltonian is invariant with respect to arbitrary symmetry operations, so that the adiabatic potential belongs to the totally symmetric representation (A-type or 2-type). Therefore, only certain combinations of QjQj or QjQjQk are allowed for symmetry coordinates and they must span the totally symmetric representation. Secondly, the interaction matrix U... 

We start with the alternating Huckel chain in Eq. (1), which simply consists of the bond order operators pd,s for nearest-neighbor hopping between sites 2n, 2n -1 and 2n, 2n + 1. Following SSH [10,11] and LHS [8], we discuss the dimerized ground state and write the linear e-ph Hamiltonian, Eq. (11), in terms of the symmetry coordinates in Eq. (16) for single- and double-bond stretches. In the = 0 subspace, we have... 

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... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

In Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

The above results mainly apply to the Longuet-Higgins E x e problem, but this historical survey would be incomplete without reference to early work on the much more challenging problems posed by threefold or higher electronic degeneracies in molecules with tetrahedral or octahedral symmetry [3]. For example, tetrahedral species, with electronic symmetry T or T2, have at least five Jahn-Teller active vibrations belonging to the representations E and T with individual coordinates (Qa,Qb) and (Qx. Qx. Q ) say. The linear terms in the nine Hamiltonian matrix elements were shown in 1957 [3] to be... 

Since Hj does not have spherical symmetry like the hydrogen atom the angular momentum operator L2 does not commute with the Hamiltonian, [L2,H] 7 0. However, Hj does have axial symmetry and therefore Lz commutes with H. The operator Lz = —ih(d/d) involves only the 0 coordinate and hence, in order to calculate the commutator, only that part of H that involves need be considered, i.e. 

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