Lagrangian classical

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

Let us now consider the following auxiliary classical mechanics problem with the classical Lagrangian... 

The scheme proposed by Car and Parinello1 in 1985 offers an attractive solution to this problem, by propagating the wave-function together with the nuclei. The ingenious idea of Car and Parinello was to include the fictitious kinetic energy term describing the wave-function motion into the classical Lagrangian ... 

The central concept of AIMD as introduced by Car and Parrinello [1] lies in the idea to treat the electronic degrees of freedom, as described by e.g. one-electron wavefunctions ipi, as dynamical classical variables. The mixed system of nuclei and electrons is then described in terms of the extended classical Lagrangian Cex. ... 

For the extended system of ions and particles C, the Lagrangian may be obtained by extension of the classical Lagrangian for ionic dynamics by means of a (fictitious) kinetic energy term due to the particles C ... 

We may retain a formal analogy with the variation of the classical Lagrangian (eqns (8.48) and (8.49)) and with the variation of the quantum Lagrangian integral operator (eqn (8.83)) by defining the functional derivative of with respect to T (and correspondingly P ) to be... 

We next give a brief derivation of the G( )-terin in the RBU Hamiltonian, which has previously not been done. The general procedure is to derive the classical Lagrangian for the constrained motions of the model and then transforming this to the Hamiltonian forin[40, 41]. Let... 

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

Feynman and Hibbs related this quantity with the classical Lagrangian L i along all possible paths connecting the point Qo to Qn ... 

Under the absence of the external forces (equivalently with free motion anyway) the above Lagrangian has to recover in the non-relativistic limit v c the classical Lagrangian limit (the kinetic energy in fact) of the free particle... 

Molecular dynamics (MD) is an application of classical mechanics using computer simulations. Good introductions can be found in many textbooks, for example the excellent book by Tuckerman [9]. In order to carry out MD, equations describing the motion of molecules are needed. These equations of motion can be derived for example from the classical Lagrangian , a function of the kinetic (K) and the potential energy (U) ... 

One way to achieve the combination of an electronic structure calculation with a classical molecular dynamics scheme is a straightforward coupling of the two approaches. For every set of nuclear coordinates, the electronic structure problem is solved and the nuclear forces are calculated via the Hellman-Feynman flieorem. The nuclei are then moved to the next position according to the laws of classical mechanics and the new forces are again calculated from a fiill electronic structure calculation. This type of ab initio molecular dynamics is often referred to as Bom-Oppenheimer dynamics . In 1985, Car and Parrinello have introduced an elegant alternative to this approach in which the electronic degrees of freedom, as described by e.g. one-electron wavefunctions I (pi), are treated as fictitious classical variables. The system is described in terms of the extended classical Lagrangian L x... 

The problem for us is therefore to derive the classical Hamiltonian function for an electron in the presence of electromagnetic fields, which is normally done from the classical Lagrangian. Hamilton s and Lagrange s generalizations of classical mechanics are essentially the same theory as Newton s formulation but are more elegant and often computationally easier to use. In our context, their importance lies in the fact that they serve as a springboard to quantum mechanics. 

The classical Lagrangian for a particle with charge q and mass m moving in a field specified by potentials 0 and A, with velocity v, is (see e.g. Slater, 1960, Vol. 1, Appendix 4)... 

The retarded potentials are discussed in detail in the references dted. Here we need only note that the classical Lagrangian for a pair of particles interacting through these potentials is given by (Darwin, 1920)... 

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