Hamiltonian nuclear kinetic energy

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

Here, t is the nuclear kinetic energy operator, and so all terms describing the electronic kinetic energy, electron-electron and electron-nuclear interactions, as well as the nuclear-nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configrrration R. 

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Both of these matrix elements are readily computed analytically (the subscript R denotes integration over the nuclear coordinates and by definition Su and S/y vanish for / / J). In Eq. (2.11), H/y is the full Hamiltonian matrix including both electronic and nuclear terms. Each matrix element of H is written as the sum of the nuclear kinetic energy (7r) and the electronic Hamiltonian (He)... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

The proof of the Hohenberg-Kohn theorem is quite straightforward. Excluding nucleus-nucleus interactions and the nuclear kinetic energy, the Hamiltonian may be written as... 

The problem still is to finding a complete set of solutions to the molecular Hamiltonian H that contains nuclear kinetic energy operators ... 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

Here Q represents a vector of normal nuclear coordinates Qu Q2>- , Tn is the nuclear kinetic energy operator, and HSF(Q) is the electronic spin-free Hamiltonian... 

The first term on the right-hand side is identical with that of Eq. (41) (since the nuclear kinetic energy cancel the Hamiltonian matrix Hrnn can be replaced by the PES matrix Vrnn, Eq. (10)). The derivatives in the second term on the right-hand side of Eq. (48) are responsible for the formation of a nuclear coordinate and momentum dependence of the density matrix. The multitude of involved coordinates and momenta, however, avoids any direct calculation of the pmn(R, / /,), and respective applications finally arrive at a computation of bundles of nuclear trajectories which try to sample the full density matrix. 

The CC Hamiltonian has been introduced in Eq. (1) (again, the nuclear kinetic energy contribution may be removed), and the coupling to the radiation field follows from Eq. (21). The nuclear coordinates are time-dependent functions determined by Newton s equations... 

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