Energy Taylor expansion

The high-temperaUire expansion could also be derived as a Taylor expansion of the free energy in powers of X about X = 0 ... 

The potential matrix elements are then obtained by making Taylor expansions around 00, using suitable zero-order diabatic potential energy functions,... 

The free energy derivatives are also related to the coefficients in a Taylor expansion of the free energy with respect to X. In the case of linear coupling, we let Eba = XdJ-a — Up,)lkT in Eq. (9) we obtain... 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

If A = 0, then H = Hq, 4/ = o md W = Eq. As the perturbation is increased from zero to a finite value, the new energy and wave function must also change continuously, and they can be written as a Taylor expansion in powers of the perturbation parameter A. 

For a homogeneous field (i.e. the field gradient and higher derivatives are zero), the total energy of a neutral moleeule may be written as a Taylor expansion. 

The change in energy for moving a nucleus can also be written as a Taylor expansion. 

The potential energy is approximated by a second-order Taylor expansion around the stationary geometry. 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

The errors in (6.20)-(6.23) are given for the exponential form of the free energy difference, and the inaccuracy in ZL4fwd and ZL4rvs can be obtained from them easily. Note that when 5e is small, (6.22) and (6.23) give the absolute systematic error in (3AA itself (through the Taylor expansion of 6e to the second order). 

Because the concept of minimum energy path is not well-defined when multiple electronic states are involved, the initial data set is simply taken as the union of points which one considers important on each of the electronic states—for example, local minima on each electronic state. The weights of each data point, Wi in Eq. (2.34), were taken to be the same on all electronic states because they only depend on the location of the data points. Hence, the difference between electronic states (V he[)ard(R)) is manifested only in the parameters of each of the Taylor expansions ... 

Dynamical variables are estimated from the gradient information generated during energy minimization. In this way velocities are derived (in one dimension) from the Taylor expansion... 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

In order to compare our approach with other approaches dealing with adiabatic corrections we perform simple model calculations for adiabatic corrections to ground state energy. We start with adiabatic Hamiltonian (32). We now perform the following approximation. We limit ourselves to finite orders of Taylor expansion of the operators H and H g We shall use similar approximation as in [25]. The diagrammatic representation of our approximate Hamiltonian will be... 

To extract a value of g0 from our calculations, it is useful to think about the functional form of Etot(a). The simplest approach is to write the total energy using a truncated Taylor expansion ... 

We will begin with a simple example, the vibrations of an isolated CO molecule. More specifically, we consider the stretching of the chemical bond between the two atoms in the molecule. For convenience, we assume that the bond is oriented along the x direction in space. The bond length is then defined by b = xc xo, where xc and x0 are the positions of the two atoms. A Taylor expansion for the energy of the molecule expanded around the equilibrium bond length, bo, gives... 

We begin by defining a set of N atoms and writing their Cartesian coordinates as a single vector with 3N components, r = (r, ..., ro,N). If locating the atoms at ro is a local minimum in the energy of the atoms, then it is convenient to define new coordinates x = r ro. The Taylor expansion of the atom s energy about the minimum at ro is, to second order,... 

The first derivatives in a Taylor expansion, similar to Eqn (28), of the energy E with respect to the occupation numbers rii provide the KS-eigenvalues, as stated by Janak s theorem, and the second derivatives ... 

Valach (1999) has proposed the use of the five-parameter eqn (Al.ll) based on a Taylor expansion of the quantum stabilization energy ... 

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