Single reference Coupled Cluster calculation had become standard
for computing highly accurate solutions of the electronic Schrodinger
equation. State speci c multi-reference CC in combination with DMRG
provides a well proved tool to compute strong correlation e ects. 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 de ne 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 de ned by P. In the spirit CC
formulation we use the ansatz k := eTk j ki 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)
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