Hamiltonian integrals

The angles a and P define the orientation of the sample relative to the Bo-field cor denotes the rotation frequency. For the REDOR reference experiment, the rotor-synchronised spin-echo experiment for the S nuclei (cf. Figure lA), the dipolar Hamiltonian integrated over one rotor period Tr averages to zero... 

The explanation is that the calculated velocity is not linear and the 4x4 matrices commute only with the momenta but not the Hamiltonian. Integration under these conditions yields the velocity as made up of two sums... 

Because of the symmetry of the Hamiltonian integrals involving ipa T) and 7>a (2) are identical and calculated only once. 

In Eq. (9) the summation is over spin. In atomic units the one- and two-electron Hamiltonian integrals are obtained by integration over electronic coordinates according to... 

The Hamiltonian integrals depend on the molecular geometry in two ways. The first is trivial and arises because the Coulomb interactions between the electrons and the nuclei depend on the geometry. The second is more complicated and arises because the orbitals are themselves functions of the geometry. The reason for this is that the MOs are expanded in a finite set of AOs fixed on the nuclear centers. A consequence of using a finite set of AOs is that we are presented with a different basis set at each geometry. 

Here the one-electron and two-electron Hamiltonian integrals are given by ... 

The one-electron Hamiltonian integrals are also parametrised. In fact, for hydrocarbons the formulae are as follows ... 

Because of the orthonormality of the spin factors the overlap and one-electron Hamiltonian integral matrices over the spin-basis rvill be diagonal blocked . Two (identical) m x m blocks on the diagonal of a 2m x 2m matrix with two m X m blocks making up the full matrix ... 

Tupper, P. A non-existence result for Hamiltonian integrators (2006). http //tirxiv.oig/abs/ math/0607641... 

D and are the unperturbed and first-order perturbed density matrices H is the one-electron Hamiltonian integral... 

If quantum nuclear fluctuations are sought, from eq.(8) one only needs tiie first term, = Kn + He(r ). These fluctuations include rotations as a whole of a frame rigidly bound to tiie stationary external Coulomb sources. The coupling terms in W, involving tiie momenta operators of electrons and nuclei, have no diagonal components tiiey will contribute to changes in the populations of stationary levels related to tiie hamiltonian. Integrating over electronic coordinates witii a particular Hn(p cc n)- the collective nuclear motion can be obtained as a solution to... 

The Hamiltonian integrals and the overlap integrals are calculated with atom-centered localized atomic orbitals under a two-center approximation. These atom-centered orbitals are constructed by solving modified KS equations of spherical pseudoatoms with confinement potentials ... 

Once these orbitals are constructed, the Hamiltonian integrals and the overlap integrals in Eq. 16.15 can be calculated for diatomic distances with grids and stored in files (Slater-Koster files), avoiding the most time-consuming SCF procedure in DFT. For SCC-DFTB, the second order term is considered, and the Hamiltonian integral has the form ... 

An isolated system is characterized by a time-independent, translationally invariant and rotationally invariant Hamiltonian. Integration of the classical equations of motion for such a system leads, in the limit of infinite sampling, to a trajectory mapping a microcanonical (Nyf ) ensemble of microstates Assuming an infinite numerical precision, this is also what a standard MD simulation will deliver. 

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