2.2 Normal ordered Vs the New Ansatz for ΩV and Ω
In the CC formulation, ΩC = exp(T), and ΩV has to be chosen properly to get an extensive expression for ∆E. We will, from now on, use a closed-shell function Φ0 as the vacuum to formulate our theory. In ΩV, or in the full Ω for open-shell energies, there will be additional cluster operators which we generically denote by S. We denote the holes by Greek letters α, β, γ etc, and the particles by Latin letters p, q, r, etc.
For the theories for energy-differences, rather than for state-energies per se, it is important to develop models of differential correlation. In such situations, CC theories treat the closed-shell ground state in terms of various n hole - n particle (nh − np) cluster operators, and the excited/ionized states of interest are described by a cluster-expanded wave-operator which include – in addition to those pertaining to the ground state – extra valence cluster operators Se involving excitations into or out of the partially filled ‘valence’ or ‘active’ orbitals (specially those inducing open-shell correlation effects), and operators Sr which bring in differential correlation and orbital relaxation effects. These latter cluster operators involve again the various n hole - n particle excitations, but additionally have excitations out of or into valence orbitals. These are thus valence cluster operators. An important class of valence cluster operators involves n hole - n particle excitations in the presence of a passive scattering of electrons between the same valence orbitals. They are cluster operators with spectator valence or active lines. The overall effective nh − np excitation amplitudes from an open-shell configuration is thus a sum of the parent ground-state nh − np amplitude and the additional amplitudes containing spectator valence orbitals. The latter ones thus bring in the relaxation of the ground state excitation amplitudes to the values they should have in the open-shell configurations, and thus bring in the differential correlation/orbital relaxation effects. In case one starts out with the set of mean-field orbitals that are optimal for the closed-shell ground state (i.e. the ground state Hartree-Fock (HF) orbitals), the 1h − 1p excitations with spectator orbitals bring in orbital relaxation effects and 2h − 2p excitations with spectator orbitals bring in the dominant differential correlation effects.
The traditional CC based correlation theories for energy differences posit on the Ω
V a normal ordered exponential [
4,
5,
6,
7,
8,
9,
12] involving the open-shell excitation operators
Se and the relaxation/differential correlation operators
Sr. The normal ordering in Ω
V ∼ {exp(
Se +
Sr)} is performed with respect to Φ
0 taken as vacuum. The valence-universality of Ω
V implies that Ω
V is the same for all the model spaces
Sm where m runs from
NV, the target
NV - valence space, all the way down to 1, the one-valence space. Owing to the normal ordering in Ω
V, there is a hierarchical decoupling of the cluster-amplitudes
Se and
Sr of different valence ranks [
9]. The use of the closedshell Φ
0 as the vacuum ensures that
Se and
Sr are spin-scalars and can be described by spin-free unitary generators [
16,
17,
18]. This makes the spin-adaptation a rather simple and straightforward exercise.
The advantages of the normal ordered Ansatz for Ω are, however, off-set somewhat by two difficulties. One is that the use of a valence-universal Ω
V implies solving for the cluster-amplitudes of
Se(
m) and
Sr(
m) of all valence ranks 1 ≤
m ≤
NV, even if we are ultimately interested in the target
NV -valence situation. This is an unnecessary exercise. The other difficulty is physically more interesting, and throws light on the limitation of a normal ordered Ansatz for Ω
V to tackle relaxation and differential correlation effects. The normal ordering in Ω
V prevents contractions between all the
S operators. As a result, the powers
Sk from Ω
V involving valence excitations with more than m orbitals for an m-valence model space
Sm automatically gives zero. However, the various
nh −
np cluster operator
Tn have all powers active in Ω
C (with 1 ≤
n ≤
NC, with
NC electrons in Φ
0); so that
nh −
np Srs should be present in the same powers. If we denote by
an arbitrary
nh −
np excitation with
l spectator orbitals, then it is physically reasonable to demand that the effective
nh −
np excitation operators in Ω
V for the open-shell situation for the m-valence
Sm should contain all the powers of each of
(1 ≤
l ≤
m) should be present. This is, however,
precluded by the very nature of the normal ordered Ω
V. For a one-valence problem as in the IP calculations, the amplitudes such as
bring in orbital relaxation and two-body correlation relaxation effects, respectively;
α is the valence hole label for a one-valence open-shell model function Φ
α. Since Ω
C generates all powers of
T1 and
T2, acting on Φ
α, with
and
, we need all powers of
amplitudes coming from Ω
V to fully take care of the orbital and pair correlation relaxation terms. However any power of
will involve destruction of more than one valence occupancy and will thus give zero by their action on Φ
α. As a result, Ω
V Φ
α is effectively just
, and misses the powers of
which are crucial when the relaxation or the differential correlations effects are large. For the core-IP, the orbital relaxation of the neutral HF orbitals is very large [
25] and the usual normal ordered exponential based VUCC methods would fail in a significant manner.
There is an earlier MRCC formulation by Mukhopadhyay and Mukherjee [
26,
27] which treats the orbital relaxations and the correlation relaxations on the same footing as in the ground state by invoking the Jeziorski and Monkhorst (JM) type of Ansatz for Ω
V [
10,
11] advocated for their valence-specific MRCC (VS-MRCC) theories for state-energies
per se. The modification consists in merely replacing the microscopic hamiltonian
H by the dressed hamiltonian
= exp(−
T )
H exp(
T ) −
Egr, with
Egr as the exact ground state energy. This so called quasi-Fock MRCC is then a method for computing energy differences. Ω
V is written as Ω
V = Σ
µ exp(
Sµ)|Φ
µ〉〈Φ
µ|, as in JM formulation and each
Sµ involves all
n-body excitations from each Φ
µ which are themselves taken as vacuum; there are thus no spectator labels. Since it is again the full exponential exp(
Sµ) which acts on each Φ
µ, the orbital relaxation and correlation effects are treated to all powers. There is, however, a big price to pay. Since in general, the functions Φ
µ will be spin-nonsinglets (they will be doublet functions for IP calculations, for example), the operators
Sµ cannot easily be chosen
in a manifestly spin-free form. The use of spin-orbital based amplitudes in
Sµ would not only proliferate the number of cluster amplitudes, they would also generally lead to spin-broken solutions. It is now well-documented that, even if
SµΦ
µ is explicitly spin-adapted, the powers (
Sµ)
kΦ
µ are not necessarily so. Thus, in practice, though the use of exp(
Sµ) solves the problem of limited inclusion of orbital and correlation relaxation effects as compared to that in VU-MRCC theories, the spin-orbital formulation and the consequent spin-contamination [
12] is a major deterrent for the quasi-Fock MRCC theories using the JM-type formalism.
What is obviously warranted is the flexibility in Ω
V of the spin-free representation of the VUMRCC approach (which implies that only a singlet type vacuum has to be adopted), and at the same time allowing the exponentiation of the
S operators in contrast to a normal ordered Ω
V. A preliminary formulation to achieve these twin goals was initiated some years ago by Mukhopadhyay
et al [
28]. In this method, the
Sr operators were allowed to contract with the spectator lines. There were several limitations of this formalism. The most important among them have been (a) the potential non-termination of the MRCC equations, since the
Sr operators could be joined in the equations in a chain-like fashion up to arbitrary powers [
22,
23]: In contrast the normal ordered exponential Ω
V or the closed-shell Ωs lead only to a finite power of cluster amplitudes since all cluster operators have to be joined to H; (b) there was no way of factorizing out the
Sr operators joined to other
Sr operators and not joined to H to lead to a more compact form of the MRCC equations. This last aspect was a direct consequence of the choice of the Ω
V in [
28] which allowed powers of
Sr, but not with the proper factors which would have offered the factorization.
The relaxation-inducing cluster expansion formalism for Ω
V we are going to discuss in this paper gets rid of the above limitations by postulating an Ansatz for the Ω
V which allows restricted contractions between the
S operators and affixes specific combinatoric factors with each such powers of contracted
S operators. The specific choice of these combinatoric factors is very crucial for us, since this leads to the generation of
finite power series in cluster amplitudes for the associated MRCC equations. The theory is very general with respect to the number of valence (active) electrons or holes present in the model spaces. In this paper, however, we should discuss explicitly only the one-valence case. We use a suitable closed-shell vacuum for defining our Ω
C and Ω
V. This leads to a manifestly spin-free form for the cluster operators
S in Ω
V. A straightforward use of this Ω
V in the Bloch equation for energy-differences leads to a potentially non-terminating series of
S operators in the MRCC equations, somewhat similar to what was obtained by Mukhopadhyay
et al [
28]. However, we show that the use of the specific combinatoric factors for the powers of contracted
S in Ω
V leads to a set of equivalent MRCC equations where all the
S operators are connected directly to the dressed hamiltonian
. This lends
a finite power-series structure to the resultant MRCC equations. Preliminary versions of the formalism for the one-valence case has already been published [
22]. A brief account of the general versions has been published already [
23].
We motivate towards our development with the example of the one-valence problem. Our model function Φ
α =
aαΦ
0 is a doublet. Spectator scatterings must have to be generally included in the spin-free choice of the cluster operators to exhaust the configurations in the
Q space which have the same orbitals but which differ in the spin functions. Thus, to incorporate the linearly independent single excitations from a hole
γ to a particle
p, we need two linearly independent amplitudes, corresponding essentially to excitations with up and down spins for orbitals
γ and
p. This can be realized by choosing the two excitation operators as
. The curly braces denotes normal ordering with respect to closed-shell Φ
0. In these operators there is a spectator scattering involving the active orbital
α in the direct and exchange modes respectively.
Another possible choice for two linearly independent excitations could have been
and
. In the present formulation we would prefer to keep the spectator scattering in the direct term, so will use
. This choice is more convenient for treating theories for energies
E and ∆
E on the same footing. The single excitation like
is of the type
Se, while operators like
are of the
Sr type. Since, in the theory of the energy differences, the overall single excitation amplitude for the excitation
γ →
p will be dictated by suitable combination of the closed-shell amplitude
and the valence amplitudes
, coming from
the effect of the
s amplitudes is to ‘relax’ the value of the closed-shell amplitude
to the value appropriate for the doublet states. This is why we label the part of
S operators containing spectators by the symbol
Sr.
As we have emphasized, the traditional normal ordered cluster Ansatz Ω
V ≡ {exp(
S)} [
6,
7,
9] does not use the full power of the exponential structure, when acting on the reference function Φ, owing to its normal ordered form. For the doublet function Φ
α, {exp(
S)} acting on Φ
α is essentially {1+
S}Φ
α. The operators {
Epα} and {
Epα} are just single excitations, which modify the orbitals. If we start out with the orbitals for the neutral vacuum state Φ0, these are not the best choice for the cation described by ΩΦ
α. However, presence of all powers of single excitations - as in an exponential - would have taken care of the orbital relaxation via the so-called Thouless Theorem [
29]. The normal ordered form {1+
S}Φ
α cannot provide such powers of excitations. Although a simple exponential choice Ω
V = exp(
S) [
4,
5], will provide exponentiation of
S via the powers, the factors (
n!)
−1 coming from the exponential are
not the proper combinatoric factors (for setting a compact power series expansion in the CC equations) when
Se operators are contracted to
Sr via spectator orbitals. The correct combinatoric factors can be ascertained on physical grounds. Each term in Ω (or Ω
V) must lead to multiple excitations via product of cluster operators such that each distinct product excitation should appear only once with a factor 1. In case contractions between
S operators are permitted, the factor (
n!)
−1 imply that all the
n S operators can be joined among themselves in all possible
n! ways. But it may not be possible for all
n S operators to be joined in such a way as to lead to same product excitation.
We want to have an Ansatz for Ω which allows contractions between operators viz. spectators, and at the same time demand that each term of the various product excitations – appearing either uncontracted to one another under the normal order or appearing as composites after contractions–should appear only once, as in the ordinary exponential without spectators. It then follows that we want to have a series for Ω in normal order where the various composites,
n in number, which appear uncontracted under the normal order should appear with a factor (
n!)
−1, corresponding to (
n!) various different ways the composites can appear. However in case there are contractions between
n operators, but only there are
fn ways of joining them, then we should attach a weight
fn−1 to the composite to ensure that the various ways of joining the
S operators in the composite leading to the same product excitations should appear only once. We note here that the commuting operators in exp(
T) realizes this automatically in the SRCC theory for closed-shells. If the operators do not commute, as for
Se and
Sr, where
Se has no spectators and
Sr has spectators, the ordinary exponential introduces unphysical weights such as (2!)
−1 for quadratic powers for the contracted composites and so on. But we can have only a composite
and not
, and hence
should appear with a factor 1.
We illustrate this with a concrete example with
φα as the model function. All the possible
S operators are shown in
Fig. 1 Let us consider the product excitations coming from an
Se as
. If we use a pure exponential for Ω
V, then the product excitation leading to double excitation
γδ →
αp would appear with a factor (2!)
−1. The operators
can contract with
only from the left via
α. But put in the reversed order, they cannot be contracted from the right. If we want to have each distinct type of product excitation to appear only once, the factor with the composite obtained by joining the operators
should just be 1.
In general then, the correct combinatoric factors would appear if we assign a factor
f −1 to a composite obtained by contracting
k Se operators to
l Sr operators, where
f is the number of possible ways of joining them together leading to composites of same excitation. With this insight, the contracted product excitation from
would have a weight of 1. Clearly no
Se operators can be joined to
Sr operators from right, and all
Sr operators need not all be joined among themselves to form the composite.
To take care of the proper factors in the composites obtained via spectator contractions, we have thus suggested recently [
22] that Ω
V should be taken to be of a combinatoric cluster expansion form:
where {{⋯}} denotes a special ordering. It allows contraction of the
S operators via spectator lines, but it assigns the appropriate combinatoric factors
f−1 to each composite.
Figure 1.
The various types of S operators for the one-hole model space Φα. (a) and (b) are Se operators, and the rest are Sr.
Figure 1.
The various types of S operators for the one-hole model space Φα. (a) and (b) are Se operators, and the rest are Sr.
From now on, we shall call all composites leading to the same excitation, but having different ordering of connectivities via spectators, as topologically equivalent. All the equivalent composites have the same ‘topological weight’, f −1. Their overall contribution to the excitation can thus be taken care of by considering only one of them with a factor of 1. It turns out that the classification of the various terms in the open-shell CC equations are best done in terms of composites of equivalent topologies.
2.3 Emergence of Strongly Connected Finite Power MRCC Equations
Let us now rewrite the left side of the eq. (7) in normal order with respect to Φ
0, using the Ansatz eq. (11) for Ω
V. It is straightforward to show that the resultant terms in normal order can be written as
where the connected composite
is obtained by joining
with various powers of
S in all possible ways, at the same time joining
Se and
Sr operators among themselves in all possible ways. The factors associated with the composites are according to the definition of {{exp(
S)}}. The various terms of all powers of
S not joined to the composites can all be grouped again to form Ω
V. Using eq. (8), the right side of eq. (7) can be written as
where
is the composite obtained by joining powers of
S with
eff in all possible ways, at the same time appropriately joining
Se and
Sr among themselves. Using the linear independence of all the operators of Ω
V in a VU theory [
4,
5,
9], it then follows that
Since all powers of
S lead to excitations out of the
P space, the closed (or the model space projection) components of both sides of the equations lead to
Inspection of the left side of eq. (15) reveals very specific modes of connectivity of the various composites appearing in it. Any
S operator in a composite joined either to another
S operator via the spectator lines only, or to the
via the spectator lines only must leave some of its inactive orbitals uncontracted, and hence cannot contribute to the closed projection. We call such type of connectivity as ‘weak connectivity’ [
22,
23]. The rest of the terms would have
S operators joined to
by at least one inactive line and, in addition may generally have contractions among themselves via spectator lines. We call these composites ‘strongly connected’ [
22,
23]. The above argument shows that
eff is strongly connected, and thus cannot have more than the powers of
S exceeding number of lines in
.
We now regroup the various terms of eq. (14) in terms of strongly connected entities. Let us consider the left side of eq. (14) first. Any general term of the left side will have several
S operators joined strongly to
(i.e. not just by spectator lines), in addition to connection among themselves via the spectator lines, and additionally we have other
S operators joined just to other
S operators (or to
) via the spectator lines. These latter are thus weakly connected. Several composites will have the same strongly connected terms, but they will differ in the ways the
S operators are joined weakly to them. We denote the various strongly connected components by
Xi, where
i distinguishes the various terms. All the composites with the same strongly connected component and the same
S operators joined weakly to this components in various ways leading to same shape may be termed as weakly connected composites of same topology. Each such term will have some
Se operators joined weakly to
Sr operators via spectator lines. The
Sr operators to which they are connected are either a part of
Xi’s (viz. they are strongly connected to
), or
Figure 2.
Connectivities of Se and Sr operators, having different factors: (a) one-body Se and Sr, (b) two-body Se and Sr and (c) one-body Se and two Sr operators. The weight factors are indicated in the parentheses.
Figure 2.
Connectivities of Se and Sr operators, having different factors: (a) one-body Se and Sr, (b) two-body Se and Sr and (c) one-body Se and two Sr operators. The weight factors are indicated in the parentheses.
they themselves are weakly connected to
Xi’s. It is interesting that the contributions of all the weakly connected composites of same topology can be written as coming from just one term in which each weakly connected
Se operator is joined to
Sr operators which are strongly connected to
, i.e. they, a part of
Xi’s, and all the weakly connected
Sr operators are joined from the right to
Xi’s via spectator lines only. The weight of this term is
f −1 where
f is the number of ways the various weakly connected
Se and
Sr operators can be arranged among themselves. This is shown in
Fig. 2. The entire term on the left side of eq. (14) can then be written as
where each
Se operator is weakly connected to
X via
Sr operators in
X, and they are not joined to each other. We have made the convention of stretching them to the left of
X without any change in their contribution. This stretching is also depicted in
Fig. 3(a). The two weakly connected
Sr operators appear only on the right of
X. [⋯]
w denotes a term joined weakly to the rest of a composite.
Figure 3.
(a) Overall contribution from three diagrams of the same topology. The skeleton in the braces is the strongly connected ‘X’ operator. (b) Moving the weakly connected Sr operator on the left of eff to its right, yielding the same contribution. The quantity in the braces is the eff.
Figure 3.
(a) Overall contribution from three diagrams of the same topology. The skeleton in the braces is the strongly connected ‘X’ operator. (b) Moving the weakly connected Sr operator on the left of eff to its right, yielding the same contribution. The quantity in the braces is the eff.
Using entirely the same reasoning on the right side of eq. (14), we have
where each
Se in [⋯]
h above is joined entirely to the
vertex, and each
Se in [⋯]
w is joined to one or more
Sr vertex which are part of
eff. The
Sr operators, originally connected weakly among themselves via spectator lines from the left of
eff are all moved to its left. This operation keeps the contribution of these terms unchanged, when they belong to the same topology. As an example, we have shown in
Fig. 3(b) one term from eq. (17) where the
Sr vertices are taken from the left to the right of the
eff vertex.
Again since all powers of
Sr in {{exp(
Sr)}} are linearly independent, it follows from eqs. (14), (16) and (17) that
‘Inverting’ the ‘exponentials’ in eq. (18) leads to
where each term in {exp(−
Se)} is joined to the
X via its
vertex, and they are not joined among themselves. The strongly connected composite
X has one set of terms
Z which are ‘external’ or ‘open’ in the sense of inducing excitations to the virtual space from the model space and another set
W which is ‘closed’. Only
W contributes to
eff. Denoting the external operators by the suffix ‘ex’, and the closed ones by ‘cl’, we have, from the
Q projection of eq. (19), the relation
The
P projection leads to
Eqs. (20) and (21) are respectively our stipulated CC equations the VU theory for the cluster amplitudes for
S and for
eff in the spin-free compact formulation. Each
m-valence component of eq. (20) should separately be equated to zero to generate the
S(
m)’s. Because of the nature of connectivity, in an
m-valence component of eq. (20) no
S(
l) with
l >
m appears, as in a normal ordered formulation.
It is clear from our formulation that the expression on the left hand of eq. (20) is finite power series in S, and since all S operators are strongly connected to the vertex of . We should emphasize again that the finite series emerges entirely due to our new Ansatz for ΩV with suitable combinatoric weights.