Eckart function

For a one-dlmenslonal barrier of arbitrary shape, numerical methods can be used to calculate the transmission probability ic (12). Before the availability of large computers, this problem was often circumvented by approximating the barrier with an Eckart function (13). The symmetric form of this, which I shetll use later. Is... 

A slightly more realistic description of the change in potential energy along the minimum energy path is provided by the following Eckart function (76) ... 

Computed average vibrational free energies for H-transfer (blue) and D-transfer (green) in wild-type DHFR. The results are fitted to an inverse Eckart function, and the locations of the transition state at 5 °C and 45 °C are indicated by the vertical lines. The Boltzmann factor of between H- and D-transfers gives the vibrational free energy contribution (dominantly zero-point effects) to the overall KIE. The reaction coordinate, ARc, is defined as the difference of the distances of the transferring proton from the donor (NADPH) and acceptor (dihydrofolate) atoms. 

Strictly speaking, the concept of itself makes no sense for a potential like the Eckart one, unless one artificially introduces Zo as the partition function of a bound initial state, which is not described by this potential. That is to say, it is reasonable to consider the combination kZo, not k alone. 

It seems as if an energy value of sufficiently high accuracy has now been found for the helium problem, but we still do not know the actual form of the corresponding exact eigenfunction. In this connection, the mean square deviation e = J — W 2 (dx) and criteria of the Eckart type (Eq. III.27) are not very informative, since s may turn out to be exceedingly small, even if trial function... 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

The ab initio calculations produce values of fiy, i.e., the components of the electronically averaged dipole moment along the x y z axes defined above. In order to calculate molecular line strengths, however, we must determine, as functions of the vibrational coordinates, the dipole moment components along the molecule-fixed axes xyz (see equation (23)) defined by Eckart and Sayvetz conditions [1]. 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

The matrix element is defined relative to two-electron wave functions of coupled momenta. If now we take into account the tensorial structure of operator (14.57), apply the Wigner-Eckart theorem to this matrix element and sum up over the appropriate projections, we have... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

These tensorial products have certain ranks with respect to the total orbital, spin and quasispin angular momenta of both shells. We shall now proceed to compute the matrix elements of such tensors relative to multi-configuration wave functions defined according to (17.56). Applying the Wigner-Eckart theorem in quasispin space to the submatrix element of tensor gives... 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

The general matrix element over the term kets obeys the reduction (integration with respect to the angular momentum functions) according to the Wigner-Eckart theorem ... 

Fig. 6.4.2 Tunneling probabilities for an Eckart barrier. The barrier height is 40 kJ/mol and the magnitudes of the imaginary frequency associated with the reaction coordinate are 1511 cm-1 (solid line), corresponding to the reaction H + H2, and 1511/ /2 cm-1 (dashed line), corresponding to the reaction D +D2. The step function is the transmission probability according to classical mechanics.

Table 6.1 Tunneling corrections as a function of temperature according to Eq. (6.40). kh is the correction factor for H + H2, where the magnitude of the imaginary frequency associated with the reaction coordinate is 1511 cm-1, kd is the correction factor for D + D2, where the magnitude of the imaginary frequency is 1511/cm-1. The tunneling probabilities are calculated for an Eckart barrier. The barrier height is Ec = 40 kJ/mol.

Recall i/fa assumes that both electrons in helium experience the same effective nuclear charge a. While this may be so in an average sense, such an approximation fails to take into account that, at a given instant, the two electrons are not likely to be equidistant from the nucleus and hence the effective nuclear charges they feel should not be the same. Taking this into consideration, C. Eckart proposed the following trial function in 1930 ... 

In systems with orbitally degenerate states, we can also exploit the Wigner-Eckart theorem for the spatial part of the wave function. Use of the WET further reduces the number of matrix elements that have to be computed explicitly. 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

Figure 11.9 Eckart potentials and wave functions used in the simulation of laser catalyst Potential parameters were A = 0 a.u., B - 6.247 a.u., / = 4.0 a.u., and m = 1060.83 a.iu P denotes E, 1+) state of text, denotes E, 2 ) state of text, and P0 denotes Eq) state" text. (Taken from Fig. 2, Ref. [308].)...

The spin term is straightforward to evaluate by the Wigner-Eckart theorem but the rotational term requires further consideration. Let us introduce the projection operator onto the complete set of rotational functions between the operators Pp (J) and 2) (< >) (the closure relationship) ... 

We shall encounter many examples of magnetic dipole spectra elsewhere in this book but note briefly here a few examples which again illustrate the importance of the Wigner-Eckart theorem in determining the selection rules. Rotational transitions in the metastable 1 Ag state of O2 provide an important example for an open shell system which does not possess an electric dipole moment [75]. The 1 Ag state arises from the presence of the two highest energy electrons in degenerate n-molecular orbitals if these orbitals are denoted 7r+1 and n j the wave functions for the 1 Ag state may be written... 

One can form an idea of the magnitude of the corrections from the one dimensional case where, for several potentials, one has exact solutions of the Schrddinger equation which make a power series development unnecessary. Bell t has calculated the penetration of Eckart s one dimensional barrier (Eig. 5a) as function of the temperature. His results for the reaction rate, the classical values, for the same conditions... 

An important use of vector coupling coefficients lies in the calculation of matrix elements of the operators in the vibronic Hamiltonian. Knowing the symmetry properties of the basis functions and of the operators, the ratio of the matrix elements can be deduced by inspection of the vector coupling coefficients. Without resorting to complicated formulae, a restricted use of the Wigner Eckart theorem may be illustrated as follows. First let us reduce Table 1 to those columns involving only the decomposition products of E symmetry (Table 2). 

The Eckart conditions play an important role in this connection. We shall discuss this in more detail below, since the arguments presented apply equally well to the treatment of nonrigid molecules. Hence, to study the basis of introducing Eckart conditions, let us for a moment go back to an earlier stage where axis conventions were not yet formulated. We recapitulate that we are looking for the conditions required in order that the atomic position coordinates, rag, can be given as unique functions of 3 N-6 internal coordinates, or equivalently stated, in order that the expansion [Eq. (3.6)] can be determined as a unique inverse of Eq. (3.5). 

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