Symmetry coupled

For the construction of a molecular couple related by a center of symmetry (/couple), only the location of the center of symmetry in the space surrounding the molecule needs to the found for the S and G strings, the molecular orientation, the distance between the center of mass and the symmetry element, as well as the screw or glide translation period, must be found. Thus, the search for the / couples is much quicker than the search for S or G strings. The search for the cell translation periods is longest for triclinic structures. In summary, crystal structures are generated as follows ... 

Adiabatic states with different electronic symmetries couple not by the radial coupling of Eq. (4) but by nonadiabatic rotational (Coriolis) coupling, Hcor For X and II states of a diatomic molecule, the Coriolis coupling matrix element is given by (see Sec. Ill)... 

Keywords Trihalide cations Pseudo Jahn-Teller effect Double minima in C2V symmetry Coupled cluster and... 

In its emphasis on the the coupling of molecular orbitals by means of coordinates for nuclear motion, OCAMS resembles treatments based on the second-order Jahn-Teller effect [23, pp. 17-25], such as that of Bader [21] and developments ensuing from it [63, 64], in which a vibrational coordinate of appropriate symmetry couples occupied and unoccupied MOs of the same molecule and determines the manner of its decomposition. 

Simple as it may seem, molecular hydrogen (H2) is in fact a sophisticated quantum-mechanical construct. In most molecules, the effects of the nuclear spin on molecular properties are negligible and thus irrelevant. This is not the case, however, for H2 and some other small symmetric molecules such as H2O, H2CO, NH3, CH3F, and so on, because of their symmetry coupled with the fundamental laws of quantum mechanics [16,17]. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

If the symmetries of the two adiabatic functions are different at Rq, then only a nuclear coordinate of appropriate symmeti can couple the PES, according to the point group of the nuclear configuration. Thus if Q are, for example, normal coordinates, xt will only span the space of the totally symmetric nuclear coordinates, while X2 will have nonzero elements only for modes of the correct symmetry. 

The eigenvalues of this mabix have the form of Eq. (68), but this time the matrix elements are given by Eqs. (84) and (85). The symmetry arguments used to determine which nuclear modes couple the states, Eq. (81), now play a cracial role in the model. Thus the linear expansion coefficients are only nonzero if the products of symmebies of the electronic states at Qq and the relevant nuclear mode contain the totally symmebic inep. As a result, on-diagonal matrix elements are only nonzero for totally symmebic nuclear coordinates and, if the elecbonic states have different symmeby, the off-diagonal elements will only... 

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

A conical intersection needs at least two nuclear degrees of freedom to form. In a ID system states of different symmetry will cross as Wy = 0 for i j and so when Wu = 0 the surfaces are degenerate. There is, however, no coupling between the states. States of the same symmetry in contrast cannot cross, as both Wij and Wu are nonzero and so the square root in Eq. (68) is always nonzero. This is the basis of the well-known non-crossing rule. 

The cyclopentadienyl radical and the cyclopentadienyl cation are two well-known Jahn-Teller problems The traditional Jahn-Teller heatment starts at the D k symmetry, and looks for the normal modes that reduce the symmetry by first-01 second-order vibronic coupling. A Longuet-Higgins treatment will search for anchors that may be used to form the proper loop. The coordinates relevant to this approach are reaction coordinates. 

The remaining combinations vanish for symmetry reasons [the operator transforms according to B (A") hreducible representation]. The nonvanishing of the off-diagonal matrix element fl+ is responsible for the coupling of the adiabatic electronic states. 

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... 

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