Eigenvalue equation condition

Consider a more general eigenvalue equation without imposing the periodic boundary condition,... 

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... 

The simultaneous optimization of the latter two points can be done by solving coupled eigenvalue equations. These equations can be derived from imposing variational conditions on the energy [2] ... 

The eigenvalue equation (Eq. 6.108) and the continuity equation (Eq. 6.105) are both first-order equations, which together demand two boundary conditions. The two required boundary conditions are the axial velocities at the boundaries, the inlet and the stagnation surface. This specification requires that the residual equations at the upper boundary do not have a direct correspondence with the dependent variables. At the stagnation surface ( / = 1), the residual equations are stated as... 

The eigenvalue equation based on the boundary condition in Eq. (123) is of the form... 

The coefficients a, (3, y and5 are obtained by imposing the Bloch conditions with periodicity A,p, the continuity conditions of the wave function and its derivative at L/2, and finally by normalization in the surface unit. The solution of the eigenvalue equation for E gives the electronic energy dispersion for the n-th subband with energy... 

The most essential properly of acoustic vibrations in a nanoparticle is the existence of minimum size-quantized frequencies corresponding to acoustic resonances of the particle. In dielectric nanociystals, the Debye model is not valid for evaluation of the PDOS if the radius of the nanociystal is less than 10 nm. The vibrational modes of a finite sphere were analyzed previously by Lamb (1882) and Tamura (1995). A stress-free boundary condition at the surface and a finiteness condition on both elastic displacements and stresses at the center are assumed. These boundary conditions yield the spheroidal modes and torsional modes, determined by the following eigenvalue equations ... 

X dx2 Y dy2 Since the left-hand side of Eq. (4.7) is a function of x only, and the right-hand side of Eq. (4.7) is a function of y only, then both sides should equal a constant. A constant, -X,2, is selected as in Eq.(4.8) such that the equation in x (with both boundary conditions homogeneous) will give an eigenvalue equation. 

The conditions Eq. 41 represent a standard m-dimensional eigenvalue problem. Since H is Hermitian, the eigenvalue equations have exactly m solutions... 

Indeed, one can analyze In the same manner the evolution of the system under consideration under conditions of reversibility of all of the elementary reactions in scheme (3.30). Unfortunately, in this situation the analytic solution of the eigenvalue equation in respect to parameter X appears unreasonably awkward. However, if the kinetic irreversibility of both nonlinear steps are a priori assumed, it is easy to find stationary valued (Y, Z ), and we come to the preceding oscillating solution. At the same time, near thermodynamic equilibrium (i.e., at R aa P), there exits only a sole and stable stationary state of the system with (Y Z R). 

Since there are electrons, there are at most terms in the expansion (4-4). The L group electrons now only enter (3-7 a) through the matrix elements of the density operator derived from their ground state wave function Vlo, pKo - i> 2)— 2) for short. The stationary value condition applied to Eq. (3-6 a) can be transformed in the usual way into a matrix eigenvalue equation for the eigenvalues Ewm and the expansion coefficients C in Eq. (4-4)... 

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. 

The Contracted Schrodinger Equation is studied here in a spin-orbital representation coupled with the S2 eigenvalue equation as an auxiliary condition. A set of new algorithms for approximating RDM s in terms of the lower order ones are reported here. These new features improve significantly the method. 

Because local-scaling transformations preserve the orthonormality of basis functions, condition (54) is immediately fulfilled. Hamiltonian orthogonality (Eq. (55)), however, is not satisfied. For this reason, one must solve the eigenvalue equation (51]... 

The eigenvalue Eqs. 34 and 35 are transcendental equations for imknown modal propagation constants. After solving the eigenvalue equations, the field profiles can be determined by substituting the values of modal propagation constants fi into the boundary conditions and calculating the amplitudes and a i for TE modes and fcf and hj for TM modes (i = 1,2,3). 

The derived EOM equations (6) [(7)] are linear matrix eigenvalue equations for the exact excitation energies (ionization potentials and electron affinities). The eigenvectors give an satisfying conditions (12) and (13) (with o 0 =0 for IPs and EAs). The most general Oj is expressed in terms of the basis operators X >[Pg.12]

We noted earlier that the matrix eigenvalue equation which determines the solutions of the Hartree-Fock equation is equivalent to the condition... 

Solve the Fock matrix eigenvalue equations given above to obtain the orbital energies and an improved occupied molecular orbital. In so doing, note that the normalization condition <0i i> = 1 = ]SCi gives the needed normalization condition for the expansion coefTicients of the 0i in the atomic orbital basis. 

Under this condition, the sum (168) in the eigenvalue equation (167) is dominated by the term with E, E , so that one obtains successively... 

As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all nuclear permutations, with and without the all-particle inversion operator, that leave the full Hamiltonian invariant. Nuclear permutations are defined here in the same way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and C are replaced by B, C, and A, respectively. The inversion operator, E, inverts the positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are found, which together form a group that is isomorphic to Ds. How is this related to our previous point group At this point it is very important to recall that the state of a molecule is not only determined by its Hamiltonian but also, and to an equal extent, by the boundary conditions. The eigenvalue equation is a differential equation that has a very extensive set of mathematical solutions, but not all these solutions are also acceptable states of the physical system. The role of the boundary conditions is to define constraints that Alter out physically unacceptable states of the system. In most cases these constraints also lead to the quantization of the energies. 

---