Cartesian coordinates, kinetic energy

Finally, we shall look briefly at the form of the non-adiabatic operators. Taking the kinetic energy operator in Cartesian form, and using mass-scaled coordinates where Ma is the nuclear mass associated with the ath... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

It is convenient to define a set of 3N mass-weighted Cartesian displacement coordinates q, q2,..., q N such that the first three q s are the components of Qi, the fourth, fifth and sixth q s are the components of Q2, and so on. The kinetic energy T can therefore be written... 

Here, A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator V2 is defined as a sum of differential operators (in cartesian coordinates)... 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The Hamiltonian models are broadly variable. Even for an isolated molecule, it is necessary to make models for the Hamiltonian - the Hamiltonian is the operator whose solutions give both the static energy and the dynamical behavior of quantum mechanical systems. In the simplest form of quantum mechanics, the Hamiltonian is the sum of kinetic and potential energies, and, in the Cartesian coordinates that are used, the Hamiltonian form is written as... 

The classical kinetic energy T is given by the usual formula involving the time derivatives (velocities) of the Cartesian coordinates x ,... 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28)... 

Here KE(1) and KE(2) are classical kinetic energy expressions for isotopomer 1 and isotopomer 2 respectively, each containing terms for the kinetic energy of each atom in each of the three coordinates. For N-atomic molecules there are three Cartesian momenta for each atom, 3N Cartesian momenta for each molecule, and consequently 3NN Cartesian momenta for the N molecule system. The integrals in the numerator and denominator can thus be written as a product of 3 integrals of the type... 

As is well known, the vibrational Hamiltonian defined in internal coordinates may be written as the sum of three different terms the kinetic energy operator, the Potential Energy Surface and the V pseudopotential [1-3]. V is a kinetic energy term that arises when the classic vibrational Hamiltonian in non-Cartesian coordinates is transformed into the quantum-mechanical operator using the Podolsky trick [4]. The determination of V is a long process which requires the calculation of the molecular geometry and the derivatives of various structural parameters. 

Notice that H is a second order differential operator in the space of the thirty-nine cartesian coordinates that describe the positions of the ten electrons and three nuclei. It is a second order operator because the momenta appear in the kinetic energy as pj2 and pa2, and the quantum mechanical operator for each momentum p = -ih dfdq is of first order. 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

In classical terms, if we use the mass-weighted Cartesian displacement coordinates, the kinetic energy of the moving nuclei isf... 

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