Tensor matrix elements

There are three important issues to consider in the numerical solution of the Redfield equation. The first is the evaluation of the Redfield tensor matrix elements I ,To obtain these matrix elements, it is necessary to have a representation of the system-bath coupling operator and of the bath Hamiltonian. Two fundamental types of models are used. First, the system-bath coupling can be described using stochastic fluctuation operators, without reference to a microscopic model. In this case, the correlation functions appearing in the formulas for parame-... 

LIF laser-induced fluorescence (/( > sqnared unit tensor matrix element... 

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

Calculation of the angular part of the matrix elements thus remains, which can be performed exactly using tensor algebra techniques based on group theory. Since the calculation of the matrix elements is not straightforward, we provide here some details on it for the interested reader. The treatment follows the procedure described in Ref. [17]. 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit... 

The parameter A, tensors A and G, molecular dipole moment D and the constants Wi are expressed in terms of one-electron matrix elements concrete expressions for the above parameters can be found in [90], and for Wd and Ay they are also given in the next sections. The results of the calculations are presented in Table 2. 

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

Figure 4. Time dependence of selected density matrix elements for a harmonic oscillator obtained using the full Redfleld tensor. The oscillator is described by < > = 100 cm-1, 7 1(1 - 0) = 2.0 ps, and 7 2(A = 1) = oo, where n denotes vibrational levels. The system is initially prepared in a superposition of levels 6 and 7. (a) p T, (b) P34 (c) poi (d) dashed line, P66 and the solid line. P77. (From Ref. 24.)...

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

When the matrix elements are calculated for states built from /-electron configurations it is always found that the constants A% (these quantities are related to the strength of crystal field) always occur with (the sharp brackets denote integration with respect to 4/ radial function). A parameters play an important role in crystal field calculations and can be used as parameters in describing the crystal field. For the lowest L S J state they can easily be determined by using the operator equivalent technique of Elliott and Stevens [545—547] and with the help of existing tables of matrix elements. Wybotjbne [548], however, feels that a better approach is to expand Vc in terms of the tensor operators,, as... 

The most straightforward way of evaluating the angular matrix elements of Eqs. (17.12) and (17.13) is to use the method of Edmonds.5 The matrix elements are evaluated as the scalar products of tensor operators operating on the wave-functions of electrons 1 and 2. Using this approach we can write Wdas... 

The exploitation of the community of the transformation properties of irreducible tensors and wave functions gives us the opportunity to deduce new relationships between the quantities considered, to further simplify the operators, already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements. Indeed, it is possible to show that the action of angular momentum operator Lf on the wave function, considered as irreducible tensor tp, may be represented in the form [86] ... 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

The elements of the theory of angular momentum and irreducible tensors presented in this chapter make a minimal set of formulas necessary when calculating the matrix elements of the operators of physical quantities for many-electron atoms and ions. They are equally suitable for both non-relativistic and relativistic approximations. More details on this issue may be found in the monographs [3, 4, 9, 11, 12, 14, 17]. 

To conclude this chapter, let us present the main formulas for sums of unit tensors, necessary for evaluation of matrix elements df the energy operator. They will be necessary in Part 5. The matrix element of any irreducible tensorial operator may be written as follows ... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

In this chapter we have found the relationship between the various operators in the second-quantization representation and irreducible tensors of the orbital and spin spaces of a shell of equivalent electrons. In subsequent chapters we shall be looking at the techniques of finding the matrix elements of these operators. 

Eigenvalues of the operators on the right sides of these relationships are given in terms of quantum numbers of the states of the lN configurations, thereby defining the values of the matrix elements of linear combinations of tensors on the left sides. 

---