Van Vleck transformation

If we compare these equations with the projection operator expansion given in equation (7.43), we find that the expressions are identical up to and including the X2 contribution but that the 7.3 term derived here corresponds not to the X3 term in the expansion (7.43) but to its symmetrised (Hermitian) form discussed at the end of section 7.2. Since the discrepancies that arise from these two different forms are of order Xs or higher, the effective Hamiltonians derived by the two methods are identical to order X3. In the literature the Van Vleck transformation is normally implemented by use of equations (7.67) to (7.70) although the X3 contribution (7.70) has often been ignored. 

In conclusion, we note that thus far we have derived matrix elements of the transformed Hamiltonian Xfor a given block in the complete matrix labelled by a particular value of rj rather than an effective Hamiltonian operating only within the subspace of the state rj. It is an easy matter to cast our results in the form of an effective Hamiltonian for any particular case since the matrix elements involved in either the commutator bracket formulation (contact transformation) or the explicit matrix element formulation (Van Vleck transformation) can always be factorised into a product of a matrix element of operators involved in X associated with the quantum number rj and a matrix element of operators that act only within the subspace levels of a given rj state, associated with the quantum number i. This follows because the basis set can be factorised as in equation (7.47). The matrix element involving the rj quantum number can then either be evaluated or included as a parameter to be determined experimentally, while the... 

To obtain the effective Hamiltonian we need to diagonalise the SMFT Hamiltonian in Floquet space. When this is not practical we should consider perturbation expansions. The van Vleck transformation [96] will be the most convenient approach in this case. The result will be an expansion of the effective Hamiltonian Heff in terms of higher-order terms with... 

At this point of the discussion the zero- and higher-order terms will not be derived explicitly, but we will return to the van Vleck approach at a later stage, where we will treat the BMFT case. Following Goldman s derivation [98,99] and Mehring s secular averaging theory [14] the result of the van Vleck transformation yields... 

This effective Hamiltonian is again not unique but can be chosen such that its eigenvalue differences are smaller than l/2o t. Maricq [100-102] and others [14, 103] have demonstrated that the Magnus expansion of the effective Hamiltonian in AHT and the van Vleck transformation approach of the Floquet Hamiltonian are equivalent. At the time points krt the Floquet solution for the propagator in Eq. 24 has the form... 

For the block diagonalisation procedure we can use the van Vleck transformation. This procedure eliminates off-diagonal blocks of the Floquet Hamiltonian modifying the diagonal blocks. A first-order transformation removes... 

In the sum above the values of n and k as well as n" and k" are not simultaneously zero. This second-order correction term must be added to the effective Hamiltonian in Eq. 88. To obtain this result it was not necessary to change defining Dp. Thus to obtain the effective Hamiltonian in the original spin Hilbert space, it is again sufficient to apply exp(+iS j ). Only when we are interested in higher-order terms is an additional van Vleck transformation with exp(+iS ) required. This is, however, outside the scope of our discussion here. 

The H , = Hoo contribution to the effective Hamiltonian Hef / contains only-scaled isotropic chemical-shift terms. The first-order correction to the effective Hamiltonian requires the evaluation of commutators between DD elements, CSA elements and cross-terms DDx CSA. We should remind ourselves that the basic justification for using the van Vleck transformation is that the off-diagonal elements of the interactions are small with respect to the differences between the diagonal elements (see Eqs. 48a and 48b). When that is the case... 

Another approach to separation of the large- and small-amplitude modes is applicable when the kinetic and potential energy coupling terms between these modes are small. In such cases, a Van Vleck transformation may be used23. The effective kinetic energy operator for the large-amplitude modes then becomes... 

Consideration of approximate solutions to Eq. (3.35) by obtaining an effective Hamiltonian by a 2nd order Van Vleck transformation led to an expression35 for /-> /+ 1 transitions given by... 

In Sect. 3 the Wilson-Howard operator is discussed as an example of application. From this it appears that the Eckart conditions39)can be inferred from arguments which are easily extended to Sayvetz conditions40 of any type. The general derivation of Hamiltonians of nonrigid molecules can then be presented in Sect. 4, and an effective semirigid rotor Hamiltonian is formed by a Van Vleck transformation. Finally Sect. 5 gives a complete example of a calculation on a specific molecule, C3. 

In relation to the Van Vleck transformation (Sect. 4.8) we recapitulate that most of the formulae applying to rigid molecules could be generalized with only small adjustments [Eqs. (4.32), (4.50), (4.51), 4.58)-(4.60), (4.63)-(4.65) and (4.69)]. This indicates that the treatment without particular complications may be extended to cover a case where the small amplitude vibrational level is degenerate. This, however, is an object for future developments. 

An understanding of observable properties is seldom trivial. Spectroscopic energy levels are, in principle, eigenvalues of an infinite matrix representation of H, which is expressed in terms of an infinite number of true de-perturbed molecular constants. In practice, this matrix is truncated and the observed molecular constants are the effective parameters that appear in a finite-dimension effective Hamiltonian. The Van Vleck transformation, so crucial for reducing H to a finite Heff, is described in Section 4.2. 

The Van Vleck transformation is an approximate block diagonalization procedure. It allows one, in effect, to throw away an infinite number of unimportant (class 2) basis functions after taking into account, through second-order nondegenerate perturbation theory, the effect of these ignorable functions on the finite number of important (class 1) functions. The Van Vleck transformation, T, is defined by... 

The approximate block diagonalization produced by the Van Vleck transformation may be understood by first illustrating how nondegenerate perturbation... 

An original derivation of the Van Vleck transformation was given by Lowdin (1951). The following derivation is adapted from Herschbach (1956) and Wollrab... 

Equations (4.2.11 and 4.2.16) completely define the Van Vleck transformation. The matrix elements in the class 1 block of H are... 

It is important to note that the summations implied in the Van Vleck transformation are over all electronic and all vibrational levels of class 2. The perturbation summations are restricted neither to only the nearest electronic state of a... 

Before concluding this section it will be useful to discuss two examples of the Van Vleck transformation centrifugal distortion in a 3n state and A-doubling in a 2n state. 

Although centrifugal distortion is not a perturbation effect, a derivation of the form of the centrifugal distortion terms in Heff provides an excellent illustration of the Van Vleck transformation. If the vibrational eigenfunctions of the nonrotating molecular potential, V(R) rather than [V(R) + J(J + 1)H2/2/j,R2, are chosen as the vibrational basis set, then the rotational constant becomes an operator,... 

The Van Vleck transformation incorporates the effect of all Av 0 matrix elements into the Av = 0 block of the 3II Hamiltonian. The perturbation summation... 

As discussed in Sections 3.5.4 and 5.5, the A-doubling in a 2II state can result from interactions with remote 2E+ and 2E states via Hrot and Hso. The Van Vleck transformation defines [Eqs. (5.5.1a) - (5.5.3a)] three second-order parameters (o,p, and q) that appear in the 2II block of IT2-. These second-order 2II 2E interaction parameters cause both e// independent level shifts as well as A-doubling. The e//-dependent terms all arise from the e//-dependence of the... 

Perturbation theory is an extremely useful analytic tool. It is almost always possible to treat a narrow range of. /-values in a multistate interaction problem by exactly diagonalizing a two-level problem after correcting, by nondegenerate perturbation theory or a Van Vleck transformation, for the effects of other nearby perturbers. Such a procedure can enable one to test for the sensitivity of the data set to the value of a specific unknown parameter. 

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