Eigenvalue equations, reduced

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

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... 

Even given a hypothetical set of necessary and sufficient Al-representability constraints, however, the solution of CSE(2) is only unique provided that the eigenvalue w is specified and fixed. Because w does not appear in the ICSEs, a unique solution of ICSE(l) and ICSE(2) is obtained only by simultaneous solution of these equations subject not only to Al-representability constraints but also subject to the constraint that w = tr(lT2 >2) remains fixed. For auxiliary constraint equations, such as the reduced eigenvalue equation for the operator... 

It must be understood that there are as many eigenvalue equations for this Hamiltonian as there are values of Q for the H-bond bridge coordinate. Thus, the meaning of the notation fl> (Q) in the ket t(Q)), is that this ket is parametrically dependent on the coordinate Q. Of course, when the H-bond bridge is at equilibrium, that is, Q = 0, the Hamiltonian involved in Eq. (28) reduces to the Hamiltonian (21). This leads us to write the following equivalence between the ket notations met, respectively, in Eqs. (24) and (28) ... 

For co = 0, this equation reduces to (14), whereas dm/dz = 0 reproduces (23). Mathematically, Eq. (24) is a well-known random-potential eigenvalue problem, which can be solved numerically or by transfer-matrix methods [5, 153],... 

In general the relaxation to equilibrium of E(t) is nonexponential, since the rate matrix in the master equation has an infinite number of (in principle) nondegenerate eigenvalues if there are an infinite number of states n). There are, however, two instances where the relaxation is approximately exponential. In the first instance one assumes that the initial nonequilibrium state has appreciable population only in the first two oscillator eigenstates, and further that k,. 0 k,, m and k0. t k0 m for m > 2. If one neglects terms involving these small rate constants, the master equation reduces to a pair of coupled rate equations for a two-level system ... 

These equations reduce the problem to that of determining the eigenfunctions and eigenvalues of a three-dimensional linear Hermitean operator F which itself involves the eigenfunctions which are to be determined. In principle, therefore, the equations can be solved by an iterative procedure in which one postulates an initial set of eigenfunctions, calculates a new set from Fock s equations, and rep>eats this process until convergence is obtained. This, however, is an extremely arduous task, even for atoms, and for molecules it is absolutely necessary to imp>ose further restrictions on the molecular orbitals y>, before it becomes even remotely practicable to determine the form of these orbitals. The most important of these further restrictions is the assumption embodied... 

By operating with (114) and (98) on (113) we obtain the radially reduced Dirac eigenvalue equations... 

To progress further toward practical implementation, specific choices must be made for how one is going to approximate the neutral molecule wave function lO, N) and at what level one is going to truncate the expansion of the operator Q K) given in Eq. (5). It is also conventional to reduce Eq. (7) to a matrix eigenvalue equation by projecting this equation onto an appropriately chosen space of A + 1-electron functions. Let us first deal with the latter issue. 

This reduces the Schrodinger equation to = 4/. To solve the Schrodinger equation it is necessary to find values of E and functions 4/ such that, when the wavefunction is operated upon by the Hamiltonian, it returns the wavefunction multiplied by the energy. The Schrodinger equation falls into the category of equations known as partial differential eigenvalue equations in which an operator acts on a function (the eigenfunction) and returns the... 

We choose D to be the unitary matrix that diagonalises e. This choice, which uses up n(n — l)/2 of the degrees of freedom in the system of equations, reduces the number of unknowns associated with the matrix e from n n + l)/2 complex numbers to n real numbers which, of course, exactly compensates (the eigenvalues of a Hermitian matrix are always real). 

In practice billions of integrals of type (12) have to be calculated, stored and reread in each iteration. Eq. (5) is a pseudo-eigenvalue equation because C has to be known in beforehand. Therefore, one starts with an assumption of C and solves Eq. (5) iteratively until convergence has been achieved. Matrix eigenvalues are mostly computed according to the method of Davidson [5]. In its brute force form the Hartree-Fock SCF method needs a computation of iV (AT = Number of functions in its basis set x(rl)). Due to some tricks one can reduce this number to With the MP2 approach about... 

The simplest truncation of the eigenvalue equation (2) for the excitation energies is to ignore all coupling between poles, except that between a singlet-triplet pair. This is equivalent to setting (gl/nxcl ) to zero, for q q. (We have dropped the spin-index on these contributions, since we deal only with closed shell systems). Then the eigenvalue problem reduces to a simple 2x2 problem, with solutions... 

The PPP CO method > is derived by taking into account explicitly, as in the case of molecules, only the n electrons of the system. Putting S(A ) = 1, equation (3.1) reduces to the matrix eigenvalue equation... 

We need to reduce this eigenvalue equation to a manageable size, so that it can be solved. To this end, we note that from the definition of the dynamical matrix... 

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