Technische Universität Berlin

Institut für Mathematik, FG Modellierung, Simulation & Optimieru

Single reference Coupled Cluster calculation had become standard

for computing highly accurate solutions of the electronic Schrodinger

equation. State specic multi-reference CC in combination with DMRG

provides a well proved tool to compute strong correlation eects. We

aim to compute also degenerate and nearly degenerate states as well by

a multi-state version of the bivariational principle, suitable for derivation

of approximate multi-state (state-universal) coupled cluster meth-

ods. There the idea is that for a Hamiltonian with n quasi-degenerate

ground states, i.e. the n lowest eigenstates, we seek the projection P

onto this set of eigenstates, that is

P =

Xn

i=1

j nih ~ nj : (1)

Indeed, we will dene our oblique projector P via a generalization of

the bivariational principle which goes as follows: Consider the function

S(P) = Trace(HP). Requiring S to be stationary upon arbitrary

variations in the projector P (i.e., variations that preserve P2 = P

and Trace(P) = n) leads to the two-sided Bloch equation

(I ?? P)HP = 0; PH(I ?? P) = 0; (2)

that is, the range of P (Py) is a right-invariant (left-invariant) subspace

of H. The value of the functional at a critical point S =

P

i Ei,

with n exact eigenvalues Ei, and He = PHP, an n n matrix, has

Ei as its eigenvalues, while its eigenvectors determine the left and

right eigenfunctions of H in the bases dened by P. In the spirit CC

formulation we use the ansatz k := eTk jki with a reference k in a

1

CAS space , Tk consists of external excitations, together with the dual

functions ~ k = h( ~k+k)je??Tk . Applying the bivariational formulation

we derive a system of equations for these unknowns, which could be

solved by a self consistent iteration.

Moreover the nal method becomes extremely closed to state speci

c CAS-CC and avoids the usual diculties like over parametrization

of other state universal CC methods. The cost of the solution of the

individual CC calculations in each iteration remains similar to that

for standard single double coupled cluster calculations, like equation of

motion (EOM). But it has to be incorporated into a self consistent iteration.

The bottle neck of the present approach remains to be the full

CI solutions on the CAS space. For this purpose we recommend recent

highly ecient full CI solvers, e.g. by tensor approximation (DMRG)

or Monte Carlo FCI.

At the end we want to discuss related convergence analysis, mostly

is based on earlier analysis of the Coupled Cluster approximation and

on joint work with F. Faulstich (Berkeley), A. Laestadius (OlsoMet)

and S. Kvaal (Dept. Chemistry U Oslo)