Hermite polynomials table

For reference, the Hermite polynomials for = 0 to = 10 are listed in Table 4.1. When needed, higher-order Hermite polynomials are most easily obtained from the recurrence relation (D.5). If only a single Hermite polynomial is wanted and the neighboring polynomials are not available, then equation (D.4) may be used. 

X-ray Charge Densities and Chemical Bonding table 2.1 Low-Order Hermite Polynomials... 

Here, the permutations of j, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). 

The first few Hermite polynomials are given in Table 3.1 their recursion relation is ... 

Berlyand, O.S. Gavrilovo, R.I. Prudnikov, A.P. Tables of integral error functions and Hermite polynomials. Oxford Pergamon Press 1962... 

Since the f actorial of a negative integer is infinite, all terms f or which 2k > n have vanishing coefficients so the series reduces to a polynomial. When n is even, the upper limit of k is n/2 when n is odd, the upper limit is n — l)/2. The first few Hermite polynomials are given in Table 21.1. 

Quantum dynamics simulations of the UV absorption spectrum and of the electronic state population dynamics of the molecule excited by a short laser pulse resonant with the transition to the bright B2u t t ) state, based on the models described in Sect. 5.3 were performed using the MCTDH method in the multi-set formalism (see Sect. 4.2.5 in Chap. 4). For the representation of the Hamiltonian and the wave function, a Hermite polynomial DVR scheme [60] was used for all the degrees of freedom. The number of SPF and primitive basis functions used in the calculations are listed in Table 5.4. Test calculations with both larger primitive and SPF bases have been... 

TABLE 8.1 The harmonic oscillator wavefunctions. The Hermite polynomials and normalization constants are given for the first six harmonic oscillator wavefunctions rjvly) =, where the... 

The normalization constants Ay for the Hermite polynomials in Table 8.1 are for unitless wavefunc-tions r y y) = AyH y)e of the unitless coordinate... 

The 0 part of the differential in equation 11.46 does have a known solution. The solution is a set of functions known as associated Legendre polynomials. (As with the Hermite polynomials, differential equations of the form in equation 11.46 had been previously studied, by the French mathematician Adrien Legendre, but for different reasons.) These polynomials, listed in Table 11.3, are functions of 0 only, but have two indices labeling the functions. One of the indices, an integer denoted , indicates the maximum power, or order, of 0 terms. (It also indicates the total order of the combination of cos 0 and sin 0 terms.) The second index, m, specifies which... 

Hermite polynomials in Table 11.2. Compare your answer with the wavefunction defined by equation 11.19. 

These identities (7 and 11-13) form a very practical system of equations for the generation of the n a "anharmonic polynomials. In Table 4 we list first few and qj s. The present identities are sli tly more complicated than those of the classical orthogonal polynomials. For example the Hermite polynomials are defined by... 

Also, generalizations of the H x) to nonintegral complex n exist, called the Hermite functions. The lowest-degree Hermite polynomials are listed in Table 6.7. 

It should be observed that in contrast to the Hermite and Legendre polynomials, the Laguerre polynomials contain both odd and even powers of x. The first few are given in Table 22.1. 

---