First Eigenvalues by Eq

Such a method was first proposed by Wyatt and co-workers.43 7,56 In their so-called recursive residue generation method (RRGM), both eigenvalues and overlaps are obtained using the Lanczos algorithm, without explicit calculation and storage of eigenvectors. In particular, the residue in Eq. [41] can be expressed as a linear combination of two residues ... 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

Equation of this form for an open-shell situation was apparently first considered by Bloch/116/, and is now known as the Bloch equation/4/. Eq.(6.1.16) indicates that the diagonalization of H, which is defined in the model space only, will furnish us with the eigenvalues E and the associated coefficients C. ... 

This equation is obtained by equalizing to zero the determinant formed by the coefficients C, Cj, C3, and C4 in Eq. (17.89) under the four boundary conditions given by Eq. (17.90). This eigenvalue equation has infinite solutions for XI. The six first values are given in the following table. 

In the EOMXCCSD(PE3) or PE3 scheme, the approximate form of the transformed Hamiltonian Hn given by Eqs. (178)-(180) is first used to construct the ground-state XCCSD equations, Eqs. (141) and (142). These do not require other than one- and two-body components of H >Qpen and we can construct them using the h la and h matrix elements corresponding to Eq. (178). The EOMXCCSD eigenvalue problem, Eqs. (150) and (151), requires, however, that we calculate the three- and four-body contributions... 

For A = 1 this is known as the augmented Hessian (AH) method. It was first proposed by Lengsfield , and used with various modifications by several authors . It can easily be proved that H —el is always positive definite if e is the eigenvalue obtained by solving Eq. (33). It can also be shown that the AH method is quadratically convergent . A value A > 1 has the effect of further reducing the step length x. In fact, as shown by Fletcher and... 

The first three terms of HROT have diagonal matrix elements exclusively. This diagonal part of HROT is the rotational energy of the JMflAST,) basis function. The eigenfunctions [defined by Eq. (2.3.40)] of the rotational eigenvalue equation,... 

There are two problems here. One is easily disposed of The gradient is zero at r = 0 (at the saddle point), so the scheme in Eq. (3.10) does not progress away from the saddle point. One can show, however [40], that a MEP must approach a stationary point along the direction of the eigenvector of the force constant matrix, F, with lowest eigenvalue. At the saddle point, there is one negative eigenvalue of F, so we can simply replace the first step in Eq. (3.10) by... 

We wish to find the bound state eigenvalues by first calculating the free-particle Green s function Gq(x, x, ), then solving the integral equation (7.17a) for G(x, x, E) and then, finally, finding the values of E for which G(x, x, E) blows up. Since Jto = Eq. (7.15) becomes... 

As a consequence if S 0, the first term of Eq. (6b) for the nondiagonal matrix element of the Hamiltonian cancels in the eigenvalue equation and bonding is governed by the second term in Eq. (6b). 

The first term in Eq. 50 mirrors the displacement (drift) of the GGS as a whole under the constant external force it involves the friction which acts on the whole GGS, foes = Motf. The second term displays the intra-GGS relaxation and is governed by the set of relaxation times (eigenvalues) of the GGS. As we will see in the following, the bead displacements given by Eq. 50 are a very useful tool in probing the dynamical features of polymer systems with complex topologies. 

The expression in Eq. 6, however, does not have the variational property given by Eq. 2 unless n is the exact fully self-consistent charge density. This is because the sum of electron eigenvalues, the first term on the right of Eq. 6, contains information on both the input charge and the output charge of a particular iterative cycle in the calculation. Thus, E otal a functional of... 

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