Harmonic molecules

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

VI. Nonlinear Optics of Polyatomic Harmonic Molecules in Condensed Phases— Eigenstate-Free Spectroscopy... 

VI. NONLINEAR OPTICS OF POLYATOMIC HARMONIC MOLECULES IN CONDENSED PHASES— EIGENSTATE-FREE ... 

The expressions derived in Sections IV and V explicitly contain four summations over molecular states (over a, b, c, and d). In Eqs. (80), (82), and (84), these summations have been rearranged, so that only two summations appear explicitly (either over a, c or over b, d). The other summations are buried in the definitions of T(t), T(t), and T(t) [Eq. (79)]. A significant reduction in computational effort may therefore be achieved if we can find an eigenstate-free procedure for the evaluation of the matrix elements of T(t), T(t), and T(t) without performing any summations. This may be done for a specific model of harmonic molecules, which will now be introduced. Consider a polyatomic harmonic molecule with N vibrational modes. Its Hamiltonian is... 

In the previous sections, we have utilized Green s function techniques to eliminate some of the summations involved in the calculations of nonlinear susceptibilities. The general expression for R(t3,t2,t1) [Eq. (49) or (60)], involves four summations over molecular states a, b, c, d. In Eq. (80) we carried out two of these summations for harmonic molecules. It should be noted that for this particular model it is possible to carry out formally all the summations involved, resulting in a closed time-domain expression for R(t3,t2,t1). This expression, however, cannot be written in terms of simple products of functions of , r2 and t3. Therefore, calculating the frequency-domain response function / via Eq. (30) requires the performing of a triple Fourier transform (rather than three one-dimensional transforms). This formula is, therefore, useful for extremely short pulses when a time-domain expression is needed. Otherwise, it is more convenient to use the expressions of Section VI, whereby only two of the four summations were carried out, but the transformation to... 

Here , is the initial phase of the r th normal modes of frequency av Say now the initial phases of the normal modes are such that the bond is stretched. The differences between the vibrational frequencies means that quite rapidly the terms in Eq. (108) will get out of synch and the bond will contract. This, however, is not IVR, although it is true that in a large harmonic molecule it may take quite a while before the same bond is again extended. 

The evaluation of the integral in Eq. (14) may be done in a number of ways. Harmonic oscillator expressions for vibrational actions can be used for weakly an-harmonic molecules. A more accurate rate method, which is most useful, is based on a Fourier series representation of the coordinates and momenta in Eq. (14). In this method, the normal coordinates and momenta are calculated as a function of time by integrating the molecule equations of motion by standard numerical integration methods. The JTs and P s are represented in a Fourier series, and then the actions of Eq. (14) are evaluated from the Fourier coefficients ... 

See, for example, C. A. Parr, A. Kuppermann, and R. N. Porter, Classical dynamics of triatomic systems Energized harmonic molecules, J. Chem. Phys. 66 2914 (1977). 

Marquardt R and Quack M 1996 Radiative excitation of the harmonic oscillator with applications to stereomutation in chiral molecules Z. Rhys. D 36 229-37... 

Rasing Th, Shen Y R, Kim M W, Valint P Jr and Bock J 1985 Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation Phys. Rev. A 31 537-9... 

Zhao X L, Ong S W and Eisenthal K B 1993 Polarization of water-molecules at a charged interface. Second harmonic studies of charged monolayers at the air/water interface Chem. Phys. Lett. 202 513-20... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

The surfaces of large molecules such as proteins cannot be represented effectively with the methods described above (e.g., SAS), However, in order to represent these surfaces, less calculation-intensive, harmonic approximation methods with SES approaches can be used [1S5]. 

The Morse function rises more steeply ihan ihe harmonic potential at short bonding distances. This difference can be important especially during molecular dynamics simulations, where thermal energy takes a molecule away from a potential minimum. ... 

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