Taylor series molecules

Under steady-state flow conditions (coherent motion), a Taylor series can be applied to describe the time-dependent position of the fluid molecules ... 

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric field present, Eq, and corrections for the dipole, polarizability, hyperpolarizability, and the like ... 

In the lowest approximation the molecular vibrations may be described as those of a harmonic oscillator. These can be derived by expanding the energy as a function of the nuclear coordinates in a Taylor series around the equilibrium geometry. For a diatomic molecule this is the intemuclear distance R. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

As foe molecule executes small-amplitude vibrations with respect to the equilibrium intenmclear distance, it is appropriate to develop the potential function in a Taylor series about that position. Thus,... 

Although the equilibrium configuration of a molecule can usually be specified, at ordinary temperatures, all of the atoms undergo oscillatory motions. The forces between the atoms in the molecule are described by a Taylor series of the intramolecular potential function in the internal coordinates. This function can then be written in the form... 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

When a molecule A is attacked by another molecule B, it will be perturbed in either its number of electrons NA or its external potential vA(r). At the very early stages of the reaction, the total electronic energy of A, EA can be expressed as a Taylor series expansion around the isolated system values NA and v jfr)... 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

Non-totally symmetric vibrations lower the symmetry of a molecule and previously forbidden bands may become allowed. The Hamiltonians considered up to now were all given for a fixed nuclear equilibrium geometry. A Taylor series expansion in the normal coordinates Q around this nuclear equilibrium geometry... 

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

The electronic energy of a molecule, ion, or radical at geometries near a stable structure can be expanded in a Taylor series in powers of displacement coordinates as was done in the preceding section of this Chapter. This expansion leads to a picture of uncoupled harmonic vibrational energy levels... 

The values of the coordinates and velocity of the centre of mass of a molecule, r and v at the time t + At are expanded as a Taylor series through terms in (At)2. Making use of these expressions for both +At and —At and Newton s equation of motion, simple manipulation yields the Verlet algorithm... 

To describe the chemical reactivity in the context of DFT, there are several global and local quantities useful to understand the charge transfer in a chemical reaction, the attack sites in a molecule, the chemical stability of a system, etc. In particular, there are processes where the spin number changes with a fixed number of electrons such processes demand the SP-DFT version [27,32]. In this approach, some natural variables are the number of electrons, N, and the spin number, Ns. The total energy changes, estimated by a Taylor series to the first order, are... 

We start by considering the origin of the dipole moment, which represents the lowest order nonzero term in a Taylor series expansion of the electrostatic potential arising from a neutral body (i.e., a molecule). For an assembly of n discrete charges, the electrostatic potential at a coordinate r may be written... 

Moreover, the number and type of stationary points is even larger in the case of a polyatomic molecule, as can be shown from the analysis of the structure of the hessian matrix of the force constants (Fy = d2 V/dRt 8Rj). Take R° as one such stationary point, and expand K(R) near R° in a Taylor series (note that grad V = 0 for s = 0). Up to quadratic terms, one gets... 

We have not failed to recognize that appropriately designed (6,0) carbon and C/B/N nanotubes may display considerably enhanced nonlinear optical activity. This term refers to the response of the dipole moment of a molecule (or the polarization of bulk material) to the oscillating electric field of electromagnetic radiation.82 85 The component of the dipole moment along an axis i in the presence of an electric field e can be represented by a Taylor series ... 

The microscopic polarization of a molecule in an external field (or the dipole moment, i.e., the positions of the charges in the molecules averaged over the molecular volume) can be expanded in a Taylor series ... 

Generally, the potential function of a polyatomic molecule can be described by a Taylor series ... 

The induced molecular dipole replaces the polarization and the local field, E, acting on the molecule is introduced in place of the macroscopic field. There are two conventions in use for defining the hyperpolarizability series one is the exact analogue of the macroscopic method (B convention), the other uses a Taylor series expansion (T convention) where a factor (l/ ) is introduced into wth order terms. The notation introduced by WRBS as been used. For a noncentros5unmetric molecule subjected to an internal field,... 

The method FUERZA [169] was developed in order to avoid this ambiguity. In this method, the force constants are defined from a tensor-based formalism as follows. For a N-atom molecule or system, the 3N components of the reaction force JF due to a displacement of the N atoms of the molecular system can be expressed exactly to second order in a Taylor series expansion as... 

Technically, self-diffusion describes the displacement of a labeled molecule in a fluid of unlabeled but otherwise identical molecules. If this motion is chaotic, the mean square displacement will eventually obey the prediction of equation 13 and one can calculate the diffusion constant Dq for motion in direction g. This particular motion is difficult to observe in real adsorption systems so that simulation becomes of particular interest here. Before reviewing the literature, it is useful to consider the mean square displacement of a particle at short time rather than in the long time diffiisional limit. In the short time limit, one can carry out a Taylor series expansion to show that, after averaging, the mean square displacement in the q th direction q = x, y, z) is [60] ... 

In 1958 Cahn and Hilliard proposed a phenomenological theory for surface and interfacial tensions that was based on a general formalism for heterogeneous systems. It has a certain analogy with the descriptions of non-uniformities in magnetic and ferro-electric domains in solids. The basic idea was that the loeal Helmholtz energy density per molecule / is expanded in a Taylor series about, the corresponding quantity in a uniform phase. Mathematically,... 

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