Lagrangian density matrices

Chan, G.K.L. Density matrix renormalisation group Lagrangians. Phys. Chem. Chem. Phys. 2008,... 

Now, in order to derive a scalar Lagrangian density, we introduce the matrix and the spinors 4>0 and 4>0 ... 

This article is organized as follows. In Section 2 ab initio molecular dynamics methods are described. Specifically, in Section 2.1 we discuss the extended Lagrangian atom-centered density matrix (ADMP) technique for simultaneous dynamics of electrons and nuclei in large clusters, and in Section 2.2 we discuss the quantum wavepacket ab initio molecular dynamics (QWAIMD) method. Simulations conducted and new insights obtained from using these approaches are discussed in Section 3 and the concluding remarks are given in Section 4. 

Here h is the one-electron Hamiltonian defined in Eq. (22), 1 is the unit matrix and 8 is the matrix of the Lagrangian multipliers we included a factor of 2 in the last term to make 8 identical with the usual orbital energies. The Fock matrix and the density matrix are defined as... 

The electronic degrees of freedom are the elements of the electronic density matrix P. The Lagrangian of the system is written ... 

In equation (66a) the density matrix elements have been transformed to the AO basis in order to avoid transforming the AO derivative integrals to the MO basis for each degree of freedom, In equation (66b) the generalized Lagrangian, L, has been introduced where ... 

In terms of the Lagrangian densities, we may calculate coupled-cluster first-order properties in the same way as for variational wave functions, contracting the density-matrix elements with the molecular integrals [17]. The Lagrangian density matrices are also known as the variational or relaxed density matrices. For a closed-shell CCSD wave function, an expression for the one-electron variational density matrix is derived in Exercise 13.5. 

The optimization is here with respect to the MO coefficients, with the density matrix in the Fock operator f kept fixed. To see the equivalence, we introduce the Lagrangian... 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

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