Lagrangian Techniques

For variationally optimized wave functions (HF or MCSCF) there is a 2n -I- 1 rule, analogous to the perturbational energy expression in Section 4.8 (eq. (4.34)) knowledge of the Hth derivative (also called the response) of the wave function is sufficient for 

The idea is to construct a Lagrange function which has the same energy as the non-variational wave function, but which is variational in all parameters. Consider for example a CL wave function, which is variational in the state coefficients (a) but not in the MO coefficients (c) (note that we employ lower case c for the MO coefficients, but capital C to denote all wave function parameters, i.e. C contains both a and c), since they are determined by the stationary condition for the HF wave function. 

The first two derivatives are zero owing to the properties of the CI and HF wave functions, eq. (10.33). The last equation is zero by virtue of the Lagrange multipliers, i.e. we choose k such that dLciJdc — 0. It may be written more explicitly as 

Note that no new operators are involved, only derivatives of the CI or HF wave function with respect to the MO coefficients. The matrix elements can thus be calculated from the same integrals as the energy itself, as discussed in Sections 3.5 and 4.2.1. 

The seeond term disappears since the Cl wave function is variational in the state coefficients, eq. (10.33). The three terms involving the derivative of the MO coefficients (dc/dX) also disappear owing to our choice of the Lagrange multipliers, eq. (10.36). If we furthermore adapt the definition that dH/dX = Pi (eq. (10.17)), the final derivative may be written as 

For variationally optimized wave functions (HF or MCSCF) there is a 2n -H 1 rule, 

Tias-sincc been generalized to cover other types of wave functions and derivatives by formulating it in terms of a Lagrange function.  

The idea is to construct a Lagrange function which has the same energy as the non-variational wave function,. but which is variational in all parameters. Consider for 

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When the Lagrangian technique is used to solve the preceding problem, the least squares estimate of the measurement errors is given by... 

The Cauchy moments have been derived in Ref. [4] for CC wavefunctions, using the time-dependent quasi-energy Lagrangian technique [I]. In Section 2.1 we recapitulate the important points of that derivation and use it in Section 2.2 to derive the CC3-specific formulas. 

Faith and Morari (1979) further develop the ideas of using dual bounding through the use of Lagrangian techniques for this problem. They describe refinements which allow one to make a good first estimate to the Lagrange multipliers (needed for the bounding) and to develop rather easily a "lower" lower bound. 

Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

The extended Lagrangian technique on which the Car-Parrinello method is based can be used also in other contexts. Whenever the forces on some atoms... 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

By contrast, the Lagrangian techniques attempt to describe the concentration statistics in terms of the statistical properties of the displacements of groups of particles released in the fluid. The mathematics of this approach is more tractable than that of the Eulerian... 

Figure 9. Calculation of Rayleigh-Taylor instability using the Lagrangian technique with automatic zone restructuring. A heavy fluid falls through a light fluid, and there is a free surface on the top.

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