Nondegenerate manifold

In terms of quantum chemistry, one needs to employ a method that can properly represent the 15 magnetic sublevels of the T2 manifold. This is, unfortunately, not the case for DFT since it is restricted to nondegenerate mono-determinantal states. Thus, the simplest method which does justice to the actual physics is the CASSCF method. 

A general asymmetric top molecule has 2 J + 1 nondegenerate eigenstates for any given total angular momentum quantum number J J+1 states belong to the (—l)"7 parity manifold and J states have parity... 

The first equation as in the nondegenerate case, can be satisfied by taking the correction to be orthogonal to the unperturbed ground state vector which is satisfied by any linear combination of vectors from the manifold related to the /-th degenerate eigenvalue. Inserting the required expansion we get ... 

The dressed states (129) group into manifolds of nondegenerate triplets unless A = and then the states +,1V), 3,1V) in each manifold are degenerate. 

Let us recall that in elementary catastrophe theory critical points of potential functions are examined. A potential function can have noncritical points, nondegenerate critical points and degenerate critical points. To degenerate critical points correspond sensitive states lying in the state variable and control parameter space in the catastrophe manifold M their... 

Consider a molecule with a ground state manifold of states (multiplet) with nonzero spin but orbitally nondegenerate. At sufficiently low temperatures this multiplet is dominating and the density operator... 

A highly important role in geometry in played by Riemannian manifolds, that is, smooth n-dimensional manifolds M supplied with a Riemannian metric It is known that setting this metric is equivalent to setting in each tangent space TxM at a point x G M a, bilinear symmetric nondegenerate positive definite scalar product which smoothly depends on the point x. V a, b G TxM is an arbitrary pair of tangent vectors, this scalar product can be written in the form... 

It is well known that a Riemannian metric can always be set on any finite-dimensional smooth manifold. If a point x is fixed on M, then in the tangent space TxM, the form (a, b) can always be reduced through a nondegenerate linear transformation to the form... 

In line with this theory, a symplectic geometry in a linear space TxM s constructed whose properties are determined by setting a skew-symmetric ( ) nondegenerate scalar product (, ). This cannot be done in all cases but only when the dimension of the manifold M is even. Then an even-dimensional space TxM transforms into a symplectic space In what follows, we shall assume for convenience that a symplectic space is modelled on a Euclidean space (on which, therefore, two forms are simultaneously given, namely, symmetric and skew-symmetric). 

Hence, the degenerate Poisson bracket constructed above can be restricted (limited) to common level surfaces M23 of the integrals /2 and f. One can check that as a result, a nondegenerate Poisson bracket, arises already on the space of the functions defined on the manifold JI/23. 

Let a nondegenerate differential 2-form o) be given on a manifold M. Then, with the help of this form one can define the Poisson bracket setting... 

Developing the ideas of Fomenko described in the present chapter, Matveev and Burmistrova [310] have recently analyzed class (5) of three-dimensional manifolds, on which there exists a smooth function g with all critical points organized into nondegenerate circles, and on which all nonsingular level surfaces of the function g are union of two-dimensional tori. Brailov and Fomenko have proved that for classes H) and (Q) we have an inverse inclusion (Q) c (H), that is, in the end, classes H) and (Q) coincide. 

It is well known that if a smooth function / with nondegenerate critical points, i.e., a Morse function, is given on a smooth manifold Q, then knowing these points and their indices allows us to say much about the topology of the manifold Q. It will be shown in the present chapter that an analogue of this theory exists also in the case where on a symplectic manifold a set of independent functions in involution is given, the number of which is equal to half the dimension of the manifold. 

Lemma 3.1.5. The manifold are symplectic manifolds with a nondegenerate closed form p, which is a projection of the form Q onto M, under the mapping p — Q. In this case p p = w. 

Lemma 5.2.1. The closed differential 2-form cj defined above is nondegenerate and therefore sets the symplectic structure on a 2n-dimensional manifold T M,... 

It is readily seen that a smooth function on a compact smooth closed manifold has a finite number of critical values if all of its critical points are isolated (then their number is finite) or if they fill nondegenerate critical submanifolds (in this case their number is also finite). The latter type of functions we have called Bott functions (Ch. 2). 

In relativistic calculations, the spinors are not necessarily so well separated, due to the spin-orbit interaction. As an example of the effect of spin-orbit interaction, we choose the atoms of group 14—C, Si, Ge, Sn, Pb—which in a nonrelativistic picture have the valence configuration np, and the ground state is np i P) in LS coupling. In a relativistic model the np manifold splits into the nondegenerate sets of npi/2 and np3/2 spinors. If we apply a simple Aufbau principle, we would end up with the state 2p i22py2 J = 0) for the relativistic ground state of C. If we expand the 2p /2 spinor into spin-orbitals, we find that this state is % and / 2p ( S), which we know... 

These examples have identified two types of catastrophe points, a distinction that arises as a corollary of a theorem on structural stability. This theorem, when used to describe structural changes in a molecular system, states that the structure associated with a particular geometry X in nuclear configuration space is structurally stable if p r X) has a finite number of cps such that (i) each cp is nondegenerate (ii) the stable and unstable manifolds of any pair of cps intersect transver-sally. The immediate consequence of this theorem is that a structural instability can be established solely through either of two mechanisms in the bifurcation mechanism the charge distribution exhibits a degenerate cp, while the conflict mechanism is characterized by the nontransversal intersection of the stable and unstable manifolds of cps in p(r X). 

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