Undetermined multipliers, method

The one-to-one correspondence between electron density and effective potential, which is proven on the basis of the constrained search formulation, suggests that the effective potential can be determined directly from the electron density. Parr and coworkers developed a procedure for determining highly accurate exchange-correlation potentials from electron densities, which are calculated by high-level ab initio correlation wavefunction theories. This procedure is called the Zhao-Morrison-Parr (ZMP) method (Zhao et al. 1994). In this method, the effective potential is given by the Lagrange undetermined multiplier method with a potential. 

There are two main methods for enforcing such constraints. One is the Penalty Function approach, the other the metlrod of Lagrange Undetermined Multipliers. 

We can combine these three into one equation by using Lagrange s method of undetermined multipliers. To do so, we multiply equation (10.22) by ft and... 

The method of Lagrange s undetermined multipliers is a useful analytical technique for dealing with problems that have equality constraints (fixed design values). Examples of the use of this technique for simple design problems are given by Stoecker (1989), Peters and Timmerhaus (1991) and Boas (1963a). 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

The variational method is then used to minimize the expectation value of total energy E = (cj) H (j)) under small variation of the ip s in (19), and subject to the normalization condition of cj) ()) H (1)) = 1. (This may be done by employing the method of Lagrangian undetermined multipliers). 

A simple way of achieving this end is by application of Lagrange s method of undetermined multipliers. Let us consider the function F, such that... 

In solving for the extremum of a general function / subject to the constraints g = constant and h = constant, we can use the Lagrange s method of undetermined multipliers. That is, we can solve for... 

A more elegant and useful method was suggested by Lagrange. The fundamental difficulty is that there are fewer variables than the number of derivative conditions. As suggested by Lagrange, we can therefore introduce new constants Ai, A2,..., Ac ( Lagrange undetermined multipliers, one for each constraint) to define a new constrained function / given by... 

SIDEBAR 5.2 ILLUSTRATION OF LAGRANGE S METHOD OF UNDETERMINED MULTIPLIERS... 

The important aspect of (13.70b) is that each pa=Pa(U, V, N) has maximal ( most probable ) character with respect to the natural control variables of S. The constrained maximization procedure to find this optimal distribution by the method of Lagrange undetermined multipliers [see Schrodinger (1949), Sidebar 13.4, for further details] is very similar to that described in Section 5.2. In particular, the pa must be maximal with respect to variations in each control variable, leading to the usual second-derivative curvature conditions such as... 

Constructing G as in Eqn. (2.41) but imposing the equilibrium condition 8CP 7-= 0 and using Lagrange s method of undetermined multipliers (2A,Aj) in order to meet the structural constraints, we obtain... 

The way Lagrange s method of undetermined multipliers is interpreted here is not conventional. The approach is described in Appendix A. To guarantee that L is independent of the set [xj], set ... 

The Lagrange Method of Undetermined Multipliers. To prove important statistical mechanical results in Chapter 5, we need the method of undetermined multipliers, due to Lagrange.42 This method can be enunciated as follows Assume that a function f(xu x. .., xn) of n variables X, x2,..., xn is subject to two auxiliary conditions ... 

The process in detail is as follows. We use what is known as Lagrange s method of undetermined multipliers, introducing constants such that the quantity W, defined by... 

Write equations for minimization of total Gibbs free energy. This step employs the method of Lagrange undetermined multipliers for minimization under constraint for a discussion of this method, refer to mathematics handbooks. As for its application to minimization of total Gibbs free energy, see Perry and Chilton [7] and Smith and Van Ness [11]. 

Hi) Constrained Maximization Method of Lagrange Undetermined Multipliers The problem of constrained maximization may be posed in its most general form as follows ... 

The solution can be obtained by application of the Lagrange method of undetermined multiplier (Sec. VII.2). 

The charges are not independent variables since there is a charge conservation constraint. In the following we constrain each molecule to be neutrtd, Qia = 0. We treat the charges undetermined multipliers to enforce the constraint. The Lagrangian is then... 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

The problem is to find the set u which minimizes G for specified T and P, subject to the constraints of the material balances. The standard solution to tliis problem is based on the method of Lagrange s undetermined multipliers. The procedure for gas-phase reactions is described as follows. 

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