Multiplier, Lagrangian

In order to minimise the energy we introduce this constraint as a Lagrangian multiplier I /I), leading to ... 

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

Then a comparison of the microscopic Eq. (48) with its macroscopic counterpart Eq. (5) allows one to identify the Lagrangian multipliers as... 

The Lagrangian multiplier / is a universal constant that is independent of the type of energy. Thus, if we can evaluate (3 for one particular energy system, we will have a value of (3 for all systems. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Inserting the perturbation and Fourier expansion of the cluster amplitudes and the Lagrangian multipliers,... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

This equation shows that second- and first-order Lagrangian multipliers are not independent, so that a specific selection of will bias Thus the choice (25) for... 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

These N equations have the appearance of eigenvalue equations, where the Lagrangian multipliers are the eigenvalues of the operator f. The have the physical interpretation of orbital energies. The Fock operator f is an effective one-electron operator defined as... 

Yang-Mills fields, 249-250, 255-257 Lagrangian multiplier, conical intersection location, 488-489, 565 Laguerre polynomials, Renner-Teller effect, triatomic molecules, 589—598 Lanczos reduction ... 

Suppose that a particular set of coefficients, (Lagrangian multipliers) satisfy this equation for any variation. With the factor —2 introduced for later convenience [93] it means that... 

By the method of Lagrangian multipliers (a and / in the following) it is found in all cases for arbitrary SNi that... 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

Sg covariance matrix of the error estimates covariance matrix of variable estimates X Lagrangian multipliers... 

The Ay are Lagrangian multipliers arising from the side conditions of Eq. (7) which maintain orbital orthonormality during the minimization process. Solution of Eq. (17) in conjunction with Eq. (7) determines simul-... 

Assuming uniform prior probabilities, we maximise S subject to these constraints. This is a standard variation problem solved by the use of Lagrangian multipliers. A numerical solution using standard variation methods gives i.p6j=. 05435, 0.07877, 0.11416, 0.16545, 0.23977, 0.34749 with an entropy of 1.61358 natural units. 

The deviation of the diagonal Lagrangian-multipliers (see Eq. 6, 7) obtained for the orbitals after the given transformation from the canonical diagonal Fock-matrix elements. 

The diagonal Lagrangian-multipliers in the case of water-dimer are depicted on Fig. 4. One can see that the difference between the diagonal Fock-matrix element and the corresponding diagonal Lagrangian-multipliers is considerably smaller for the... 

In a convenient method, due to Hamilton (1964), the Lagrangian multipliers representing the constraint are algebraically eliminated from the least-squares expressions. The linear constraints are defined as... 

The constraint is enforced by introducing a Lagrangian multiplier X in the minimization function given by... 

To maximize In W, subject to the above constraints, it is convenient to use the Lagrangian multiplier method ... 

Constraints may be imposed on a set of simultaneous linear equations by the method of Lagrangian multipliers. Let the Lagrangian multipliers be — - Therefore, add to equation (A.28) the quantity... 

In deriving equation ( .35) from equation (A.34), we have made use of the fact that the indices of the sums are arbitrary and have switched a and b in the second terms of the last three lines of equation (A.34). We have also adopted without proof the hermiticity of the Lagrangian multipliers, that is, = eja. Canceling the 2 s and collecting terms yield the result... 

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