At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation.

The secular equations of Rayleigh wave propagation in partially saturated soil are derived based on the Biot's theory of wave propagation in porous two-phase media. And the saturation effects on dispersion characteristic and displacement distribution of Rayleigh wave in partially saturated soil are discussed.

For the nearly saturated soil,by analyzing the dynamic equation of nearly saturated soil,which based on mixture theory,the secular equations of Rayleigh waves in a nearly saturated half-space soil are developed.

The magnitutes of the para-meters of Coulomb interaction and internal crystal field for BGO∶Cr under O_hcrystal-field were calculated by the secular equations of 3 d~3 electrons. The absorp-tion spectrum of Cr~(3+) ions in BGO consists of three bands transited from the grou-nd state ~4A_2 to ~2E,~4T_2 and ~4T_1 states.

Group theory is here combined With the graph theory. of molecularorbitals to study the problem of solving the secular equations for highersymmetry conjugated molecules.

This paper prescnts the calculations of the 3d energy level which are split by the crystal field(CF)based on the quantun theory and solving the parameters of crystal field and secular equations.

根据量子力学理论,经过计算晶场系数,解久期方程,得到了 Ni~(2+)在 M 型钡铁氧体中取代不同晶座后,于晶场作用下的劈裂能级.

The normal coordinates are found by solving the reduced secular equations corresponding to the expression of the kinetic and potential energies of the cluster written in the symmetry coordinates.

The secular equations yield all of the thermodynamic functions for finite systems and the beginning terms of activity expansions for all of the eigenvalues of the infinite system.

The secular equations for finite tori of lattice sites are obtained by computer expansion of determinants for a hard-particle lattice gas.

Lattice gas activity series from secular equations

The second stage uses the complete basis to construct linear trial functions and to formulate the variational problem in terms of secular equations, yielding the successive vibrational and rotational states.

Although James and Coolidge (1933) solved the molecular hydrogen problem in almost complete agreement with experiment by using a 13-term 2-electron eigenfunction, his method can hardly be applied to more complex molecules. For this and other reasons (Coulson, 1938), it is still desirable to obtain a good one-electron eigenfunction, i.e., molecular orbital, for the hydrogen molecule. The best molecular orbital treatment available in the literature was given by Coulson (1938), who used a trial eigenfunction in...

Although James and Coolidge (1933) solved the molecular hydrogen problem in almost complete agreement with experiment by using a 13-term 2-electron eigenfunction, his method can hardly be applied to more complex molecules. For this and other reasons (Coulson, 1938), it is still desirable to obtain a good one-electron eigenfunction, i.e., molecular orbital, for the hydrogen molecule. The best molecular orbital treatment available in the literature was given by Coulson (1938), who used a trial eigenfunction in elliptical coordinates involving 5 parameters and obtained 3.603 eV for the binding energy of H_2, which is to be compared with the ex- perimental value of 4.72 eV. In the present investigation we have proposed a new type of trial eigenfunction for the molecular orbital: (1) with p = centers a, b, g, c, d,…… i = electron 1 or 2 (2) where the p's are centers along the bond axis a-b (Fig. 1). In this simple problem both the Fock and Hartree methods yield the same result. The molecular orbital ψ must satisfy the following integral equation: (3) where ε is the energy of the molecular orbital, F is the Fock operator which is equal to H+G(1), while H is the one-electron Hamiltonian operator: H = -1/2▽~2-1/r_a-1/r_b (4) and G(1) is the interaction potential (5) Substituting (1) into (3), we obtain the linear combination coefficients c_p, which must satisfy the following secular equation: (6) where is the solution of the secular determinant and The F_(pq)'s are not at first known, but depend upon the c_p's. A method of successive approximation must therefore be adopted. A set of c_p values may be assumed, the F_(pq)'s calculated, the secular determinant (7) solved, and a new set of c_p values found. This process is repeated until a "self-consistent" set of c_p values is obtained. The above procedure was first proposed by Roothaan (1951), not for H_2 but for more complex molecules. It was called by him the "LCAO SCF (linear combination of atomic orbitals self-consistent field) method". The new feature of the present investigation is that we not only use LCAO but also LCNAO (linear combination of non-atomic orbitals, such as x_g, x_c, x_d, …). The order of secular determinant (7) may be reduced to half if we replace the eigen- functions x_a, x_b .... by their symmetrical and anti-symmetrical linear combinations x_a + x_b and x_a-x_b. Numerical calculations have been carried out both for the three- and the two-centered molecular orbitals. The three-centered molecular orbital is (10) (11) where S_(ab) and S(ag) are the overlapping integrals between x_a and x_b, and between x_a and x_g respectively. The parameters a and g have 'been obtained to give minimum energy by the method described above. They are a=l.190, g=0.22, and the binding energy is 3.598 eV, which is almost as good as that obtained by Coulson (3.603 eV) using a trial function of 5 parameters. The two-centered molecular orbital is (12) (13) which gives a maximum binding energy of 3.630 eV for a=1.190 and R~(ac)=R(bd)=0.105 (Fig. 1). This result is 'better than Coulson's. If we allow different values for the ex-ponent α in x_a and x_g in equation (11), or if we use a four-centered molecular orbital, such as ψ=a(x_a + x_b) + b(x_c + x_d) with four parameters, namely α_a=α_b, α_c=α_d, R_(ac)=R_(bd) and the ratio b/a, it is possible to obtain a still better result. Extension of the present method to the treatment of more complex molecules is now under investigation.

It is shown that the RP-HRP (random phase and higher random phase) approximation may be derived by a variational method. The reason why the secular equation of the RP-HRP approximation is nonhermitian is then made clear and a procedure which can lead it to be hermitian is suggested. It is further pointed out that the variation principle may also serve as a method to determine the average occupation numbers

Energy matrix elements are, calculated in weak field coupling, scheme by using the method proposed in the first paper of this serles. Secular equations are solved by electronic computer. Experimental data of optical spectra of trivalent cobalt complexes are analyzed and the value of parameters △, B and C are determined. All the results are tabulated.