The Van Vleck Transformation and Effective Hamiltonians

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 4.2.13) allow specific requirements to be built into S so that T transforms H into the approximately block diagonal form desired for H. For clarity, class 1 and 2 basis functions will be denoted by Roman (a, b) and Greek (a, / ) letters, respectively. The goal is to make all first-order class l class 2 matrix elements of H vanish, 

Equations (4.2.14 and 4.2.15) cause the class 1 blocks of H and S (hence T) to commute (also the class 2 blocks), hence the only effect of T on the class 1 block of H is to fold into it some information from the off-diagonal Hi 2 block. Equation (4.2.16) forces the only nonzero elements of the second term in Eq. (4.2.12), i(H(°)S — SH ), to cancel exactly all interblock elements of Hd). Thus the lowest-order interblock matrix elements of H occur in the second-order term, and may be neglected. 

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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. 

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... 

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... 

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. 

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... 

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

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. 

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. 

An effective Hamiltonian is profoundly different from an exact Hamiltonian. This is a reason for imperfect communication between experimentalists and ab initio theorists. The two communities use the same symbols and language to refer to often quite different molecular properties. The main difference between effective and exact Hamiltonians is that the molecule gives experimentalists an empirical basis set that has been prediagonalized implicitly to account for the infinite number of remote perturbers . This is the Van Vleck or contact transformation, but it is performed by the molecule, not by a graduate student. The basis set is truncated and the dynamics occurs in a reduced-dimension state space. 

In Eq. (2.38), A, B, and C are the rotational constants in the PAS and H d the usual centrifugal distortion Hamiltonian. The main approximation made in the PAM is that the cross term -2FpP is considered as a perturbation which can be handled by successive Van Vleck transformations. The transformed Hamiltonian matrix can then be factored into smaller effective rotational matrices, one for... 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

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