On the basis of AM1 and ZINDO, according to the sum over state expression, we devised the program for calculating the nonlinear second order optical susceptibilities β ijk and β μ . The structure, electron spectra and β(-2ω, ω, ω), β (0, 0, 0) of a series of novel pull push polycyclic conjugated molecules have been studied.

On the basis of ZINDO, according to the sum over state expression, we designed the program for calculating the first hyperpolarizability β . The structures, electron spectra and β (-2ω,ω,ω), β (0,0,0) of a series of babituric acid derivative molecules have been studied.

The first hyperpolarizabilities β(-2ω;ω,ω) and β(-ω;ω,0) of hemicyanine derivatives chromophore, 4-N,N-dimethylamino-4'-N'-methyl stilbazolium (DAS), are calculated by using time-dependent Hartree-Fock (TDHF) and sum over states (SOS) method at infrared wavelength, respectively.

On the basis of ZINDO methods, according to the Sum Over States (SOS) formula, the program for the calculation of second order optical susceptibility β has been devised. The MO interaction in (dithiophene 3,2 b:3,2 d cyclopentan 4 ylidene)fullerene 60 has been studied, moreover, the electronic spectrum and second order optical susceptibility of this molecule have been calculated. The former is in good accordence with the experimental result, and the latter is a theoretical prediction.

The proofs are based on the estimate of certain character sum over F_q. If t is the period of the sequences, then the bound of the discrepancy is O(t~(-1/4)q~(1/8+ε) log q) for any ε> 0 when t≥q~(1/2+2ε). It shows that the sequences are asymptotically uniformly distributed.

We also show that with the system decohering to itspointer states, the geometric phase factor tends to a sum over the phase factors pertainingto the pointer states.

The partition function, also called the sum over states, It plays an important role in the evaluation of thermodynamic functions and in the derivation of distribution functions.

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron...

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron or many-electron bond func- tions.) Then the stationary state of the molecule is represented by the follow- ing wave function Ψ, where the summation is over all permutations of 1,2,……,2n except those within the interior of the functions,since each ψ_i is already anti-symmetrical.Obviously (2~n/((2n)/!))~(1/2) is the normalization factor. By quantum mechanics the energy of the molecule equals (1) here H_i,T_(ij) and S_(11)' are respectively the following three kinds of operators, (2) (3) (4) The third term of equation (1) is the exchange integral of electrons 1 and 1', while (1,2') is that of electrons 1 and 2'.According to the definition of bond functions,ψ_i may be written as (5) Substituting equation (5) into equation (1) and carrying out the integration over spin coordinates,we obtain (6) It can be easily seen from equation (6) that the combining energy of a mole- cule consists of two parts,one being the binding energy of the bonds represent- ed by the first term of equation (6),and the other being the interaction energy of the bonds denoted by the second term of that equation. If we choose certain functions φ_i~('s) involving several parameters and substi- tute them into equation (6),we may determine the values of those parameters by means of the variation principle. For the discussion of bond interaction energies,we develop a new method for the evaluation of certain types of three-center and four-center integrals.The interaction energy of a unit positive charge and an electron cloud of cylindrical- symmetry distribution may be written as (7) where (8) and R_0~2=a~2+b~2+c~2 The interaction energy of two electron clouds both of cylindrical-symmetry distributions with respect to their own respective axes is evaluated to be (9) (10) where is to sum over j from zero to the lesser value of n-2i and m, is to sum over i from zero to the integral one of n/2 and (n-1)/2,and is to sum over all cases satisfying the relation =m-j,while b_(n,n-2i) represents the coefficient of x~(n-2i) in the n th Legendre polynomial.

The quantum theory of two-level atom single-mode laser system of Sargent, Scully and Lamb is extended to two-mode laser in a three-level atomic system with common lower level, and master equation is obtained for the system with all the three levels being pumped. There appear four terms in the master equation that are absent in the single-mode case and those interpreted as 2-photon processes play an especial role in the present case. The master equation is represented by a probability flow diagram in two-dimensions...

The quantum theory of two-level atom single-mode laser system of Sargent, Scully and Lamb is extended to two-mode laser in a three-level atomic system with common lower level, and master equation is obtained for the system with all the three levels being pumped. There appear four terms in the master equation that are absent in the single-mode case and those interpreted as 2-photon processes play an especial role in the present case. The master equation is represented by a probability flow diagram in two-dimensions with the photon numbers n1 and n2 of the two-mode as variables. The probability flow diagram can be extended to infinity, and each arrow of it represents a term of the right hand side of master equation. By taking sum over one of the variables n1 or n2, the master equation is reduced to two equations, each of them can be represented by an one-dimensional probability flow diagram. The equation of motion under steady state is obtained by considering the correspondence between the macroscopic balance and the microscopic balance.A parameter H is introduced to reduce the equation of motion and to facilitate the mathematical procedure and a formal solution is thus obtained. Using the equation of motion and its formal solution various operation characteristics are discussed. Among them are: the threshold condition for each mode; the condition of only one mode operation; the variation of photon statistics between single-mode operation and two-mode operation and the variation of photon statistics with lower level being pumped and not pumped.

A general foimula for angular integrations in many—dimensional spaces (derived in a previous paper) is applied fo several problems conectes with solution of the Schrodinger eqation for many-particle systems.Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functims of the generaliyed radius multiplied by polynonials in the m coordinntes. The theory of hyperspherical harmonics is reviewed, and a sum rule is derived relating the sum over all the harmonics...

A general foimula for angular integrations in many—dimensional spaces (derived in a previous paper) is applied fo several problems conectes with solution of the Schrodinger eqation for many-particle systems.Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functims of the generaliyed radius multiplied by polynonials in the m coordinntes. The theory of hyperspherical harmonics is reviewed, and a sum rule is derived relating the sum over all the harmonics belong to a Particular eigenvalue of angular momen turn to the Gegenbauer polynomial corresponding to that eigenvalue.A formula is derived for projecting out the component of an arbitrary funcfion corresponding fo a particular eigenvalue of generaliyed angular momentum, and the formula is applied fo polynomials in the m coordinafes An expansion is derived for expressing a many-dimensional plane wave in terms.of hyperspherical harmonics and functions which might be called "hyperspherical Bessel functions" .It is shown how this expansion may be used fo calculate many-dimensional Fourier transforms. A fonmula is is derived expressing the effect of a group - theoretical projection operator acting on a many—dimensional plane wave. Finally the techniques mentioned above are used to expand the Coulomb potential of a many-particle system in terms of Gegenbauer Polynomials.