# State Selective Equation of Motion Coupled Cluster Theory: Some Preliminary Results

## Abstract

**:**

_{2}and F

_{2}molecules and investigate the behaviour of a number of closely related variants within the same general framework.

## I. Introduction

_{3}cation [39], most of which have significant multireference character. The NO

_{3}

^{-}anion is well described by CCSD and this is the state that is used as a reference, while the final diagonalization is over states that remove two electrons from the reference, e.g. $\widehat{i}\widehat{j}|{\Phi}_{0}\rangle $. For this reason the method is referred to as the double-ionization potential STEOM (DIP-STEOM) approach. Beyond computational efficiency a strong feature of the FSCC and STEOM approaches is the fact that they are rigorously spin-adapted and size-consistent. A weakness is that the reference state may be far removed from the actual states of interest and orbital relaxation effects may be substantial which are difficult to describe through configuration mixing. The reference state can become entirely artificial in these schemes as exemplified by results for the vibrational frequencies of ozone using the DIP-STEOM approach based on the di-anion of O

_{3}[30]. In reality the dianion of ozone is non-existent and the bound reference state is an artifact of the basis set. Therefore in such cases the DIP approach can only give reasonable (and even excellent) results in small basis sets, lacking diffuse character in particular.

_{2}and F

_{2}, are used as examples, and they share the property that qualitatively correct wave functions (for ground and valence excited states) can be described by creating two holes in a formal closed-shell di-anion determinant. Therefore our basic formulation follows a double ionization type of strategy. We will use the same parameterization for the wave function as is used in the double ionization variant of equation-of-motion coupled cluster theory (DIP-EOMCC [39]). The equations that determine these parameters are quite different however. In principle all parameters are determined specifically for the state of interest. We will discuss a number of slightly different variants and generalize the state-specific scheme such that coefficients can be determined in a state-averaged way. Finally we come full circle and investigate if the state specific coefficients for one particular state could also be used for other states, implying that they would again be transferable or valence universal. The approaches we discuss in this paper are steps towards a completely general internally contracted multireference coupled cluster approach along the lines discussed in [29,42,43]. However, none of the approaches we discuss here are rigorously size-consistent, in the sense that an open-shell system does not separate quite correctly into open-shell subsystems. For this reason we term our approaches state-selective variants of equation-of-motion coupled cluster theory, rather than multireference CC. The methods are size-intensive in the sense that the energies scale properly if closed-shell systems are added at infinity.

## II. Theory

_{1}-equation) to define Brueckner orbitals.

_{2}-equations as

_{1}coefficients.

_{2}-equations (Eqn. 7) define the Complete State Selective EOMCC approach (CSS-EOMCC), while we use the acronym RSS-EOMCC if instead of $\widehat{C}$ we use $\widehat{R}$ (Eqn. 9). As a mnemonic the acronyms contain the respective operators $\widehat{C}$ or $\widehat{R}$, while in addition the abreviation RSS-EOMCC stands for "Restricted (or Reference) State Selective EOMCC". While in the CSS-EOMCC approach $\widehat{R}={\widehat{C}}_{1}$, in the RSS-EOMCC approach we take an additional short-cut by obtaining $\widehat{R}$ from a diagonalization of the bare Hamiltonian over the 2h configurations only (DIP-TDA), and hence $\widehat{R}$ is not relaxed in the presence of dynamical correlation effects. Both of these methods are initial trial versions and in this paper we simply want to investigate the sensitivity of the results to such minor variations on the theme. There are a number of details that require further discussion. Let me briefly mention the salient points and defer the full details to a future exposition. In due time I hope to establish a single robust approach in which these details are specified once and for all. It does not serve much of a purpose to invent a large set of slightly different variants except in the testing stage.

#### Spin-adaptation.

#### Elimination of near-singularities.

**U**

_{λ}and eigenvalues λ, some of which may be small (smaller than 0.01 say). Defining projectors on the regularized subspaces $\mathbf{P}={\displaystyle \sum _{\lambda l\mathrm{arg}e}{\mathbf{U}}_{\lambda}{\mathbf{U}}_{\lambda}^{\u2020}}$ we solve the regularized equations

#### Size consistency, size-extensivity and size-intensivity.

#### Automated Implementation.

_{2}manifold) are quite involved. When we started this work we were quite aware that we needed to be able to explore various alternatives for a possible MRCC theory. Following earlier work in this direction, in particular by Janssen and Schaefer [46] and Li and Paldus [47] we built a tool to aid the derivation and implementation of many-body methodology using automation. The Automatic Program Generator (APG) we developed, generates an efficient Fortran procedure to perform the calculation starting from simple input equations like the ones used in this paper. Some details can be found in our earlier paper on a MRCC theory for doublet states [43]. The APG derives the detailed equations using essentially Wick's theorem and some further embellishments (e.g. taking symbolic derivatives, multiplying by density matrices, spin-integration). The resulting equations consist of a large number of terms (on the order of 300 spin-adapted Goldstone diagrams), consisting of multiplications of typically 3 or 4 operators. In the next step the equations are factorized so that each elementary step in the factual calculation consists of addition or contraction of two matrices. Optimal factorization constitutes an n-p complete problem in mathematics, and at present we appear to have developed a reasonable strategy to tackle the factorization issue. Finally the APG generates a Fortran subroutine that calculates the residuals of Eqns. (7-9) given a set of input amplitudes, the relevant density matrices of the reference state, and Hamiltonian matrix elements. The generated Fortran subroutines use a library of handwritten subroutines that are computationally efficient. This efficiency is illustrated by the fact that the same subroutines have been used in STEOM-CCSD calculations on free base porphyrin in a DZP+diffuse basis set [48]. The APG is well tested [49,50] and can be expected to provide fairly efficient and error-free implementations of highly advanced electronic structure methods.

#### An additional factorization step.

_{2}equation (Eqn. (7)), not in the final diagonalization. The reason to use this factorization step is that we can then avoid the use of three-particle intermediates in any of the factorized equations, and we can use unitary group operators throughout. Let me emphasize here that this step has a purely computational origin, and it is essential to employ this factorization in conjunction with the present version of the APG. We do introduce an active space in this step, but this is only done to fit the true ${\widehat{C}}_{2}$ operator. The factorization is a computational device, rather than a theoretical construction, and if the CSS-EOMCC scheme were to turn out our final method of choice we can eliminate this factorization step with some additional work. In practice we solve a regularized form of the following linear equation

_{2}amplitudes, and hence the slight error we make in the above factorization is presumably of little importance. The choice of active space is a purely computational matter, and we simple choose it to optimize the fit. It has nothing to do with the multiconfigurational problem.

#### Truncation of singles.

_{1}-amplitudes are always small (0.02 or less).

#### State averaged Calculations.

_{x}and Π

_{y}states in a diatomic for example. In a state averaged calculation only the equations for the t-amplitudes are modified:

_{λ}in the sum over included states, although in practice we always keep them equal (essential to keep degeneracies). The implementation of the programs is flexible and the subroutines written by the APG take the state-averaged density matrices on input.

#### Brueckner MCSCF

_{1}amplitudes are used to define a rotation of the orbitals defining $|{\Phi}_{0}\rangle $ and the usual Brueckner scheme is used to iterate to convergence $({\widehat{T}}_{1}=0)$.

#### Iterative Sequence of the Solution Algorithm

- Calculate $\widehat{R}$, $|{\Phi}_{0}\rangle $.
- Solve the RSS-EOMCC equations for t
_{1}and t_{2}amplitudes, using DIIS [53] to accelerate convergence. - Calculate appropriate matrix elements of $\widehat{\overline{H}}$ (in particular the numerous ${\widehat{\overline{H}}}_{aaia}$ and ${\widehat{\overline{H}}}_{aaia}$ elements are not needed).
- Diagonalize $\widehat{\overline{H}}$ over the 2h and 3h1p configurations. We can easily find multiple states in this scheme. The operator $\widehat{T}$ is defined in a state selective fashion, but if the t-amplitudes are assumed to be transferable, a number of states can be calculated simultaneously. This latter scheme is referred to as t(ransfer)-RSS-EOMCC.

- Calculate $\widehat{R}$, $|{\Phi}_{0}\rangle $.
- Iterate: [ Calculate appropriate elements of $\widehat{\overline{H}}$; Diagonalize over 2h and 3h1p configurations to obtain $\widehat{C}$; Fit ${\widehat{C}}_{2}$ to $\widehat{S}{\widehat{C}}_{1}$; Calculate residuals in the ${\widehat{T}}_{1}$ and ${\widehat{T}}_{2}$ equations and update the t-amplitudes, using $\widehat{R}={\widehat{C}}_{1}$]. In our DIIS extrapolation we use both the $\widehat{C}$ and $\widehat{T}$ operators (and orbital rotation parameters in a Brueckner calculation following reference [54]).
- In a final diagonalization of $\widehat{\overline{H}}$ we can include more states and in this way gain insight into the transferability of t-amplitudes. Such calculations are acronymed t-CSS-EOMCC.

_{2}-equation is far more involved due to the presence of ${\widehat{C}}_{2}$ (in the guise of $\widehat{S}{\widehat{C}}_{1}$).

## III. Results

_{2}and F

_{2}. In both cases we used the cc-pVTZ basis set [55], and we compare a variety of coupled cluster based methods.

#### III.a Results for O_{2}.

_{g}spin-orbitals, and the mixing between the 6 determinants leads to 3 electronic multiplets ${}_{}^{3}{\sum}_{g}^{-}$,

^{1}Δ

_{g}, ${}_{}^{1}{\sum}_{g}^{+}$. Only the triplet ground state can be accessed by conventional single reference methods and the CCSD(T) results in the cc-pVTZ basis set is seen to compare quite satisfactorily to experiment [56] (see table I). We have also included DIP-STEOM and DIP-EOM results, and we find that in particular the DIP-STEOM results are very satisfactory. In the DIP calculations we use the orbitals of the di-anion to build the reference state. This works satisfactory in the cc-pVTZ basis but results would deteriorate sharply if more diffuse orbitals are included in the basis set. In DIP-STEOM calculations "triple excitations" (4h2p type of configurations) are included implicitly and this presumably explains the slight edge that DIP-STEOM has over DIP-EOM. In table I & table II we included four variants of State Selective EOMCC results. In the t-RSS and t-CSS results we optimize the t-amplitudes for the ground state triplet and then use these amplitudes to calculate all three excited states of interest through diagonalization of $\widehat{\overline{H}}$. The vacuum determinant is built from triplet ROHF orbitals here. In all of the reported SS-EOMCC calculations the

^{1}Δ

_{g}state is obtained from a state averaged calculation that includes both partners of the multiplet. This avoids symmetry breaking. Analyzing the results, we find that the difference between the RSS and t-RSS results is completely negligible, and this indicates that the t-amplitudes are to a large extent transferable between these states that are similar in character of course. This conclusion does not hold very well, however, for the CSS and t-CSS results which exhibit significant differences. From a formal point of view the CSS results are always preferable over t-CSS of course, as all amplitudes in CSS are optimized for the state of interest. The difference between t-CSS and CSS is a little surprising however and it may signal that something is not quite stable in the CSS scheme. For the triplet ground state the only difference is the use of Brueckner orbitals in the RSS and CSS schemes, vs. ROHF orbitals in the t-RSS and t-SCC schemes. This does not make a significant difference. As far as geometries and frequencies are concerned the DIP, CSS, RSS and t-RSS methods perform about equally well, while t-CSS-EOM-CC is relatively worse.

_{g}spin-orbitals. There is a substantial admixing from a doubly excited ${\pi}_{u}^{2}\to {\pi}_{g}^{2}$ configuration in all of these states which is included in the reference state and the projection manifold in our DIP and SS based calculations, but not in MR-BWCC. We think that the choice of reference space is the dominant reason for the improvement of the present results over MR-BWCC, and the results probably do not reflect intrinsic differences in methodology. A further comparison is made with results from MR-AQCC calculations [58] in the cc-pVTZ basis set, where the 1s core orbitals were dropped from the calculation. The active space in the MR-AQCC treatment comprised the full valence space. The results for the triplet can be compared to dropped core ROHF-CCSD(T) and even ROHF-CCSDT calculations and we find that the MR-AQCC equilibrium bond distance deviates by about 0.5 pm. The effect of dropping the core is about 0.35 pm, and we find that for the triplet state the bond distance in the dropped-core MR-AQCC deviates by about 0.9 pm from the full CCSD(T) result. As seen in the table the difference between CCSD(T) and CCSDT is negligible, so we think the ROHF-CCSD(T) is a suitable reference point. For the other states the difference between our SS-EOM results and the MR-AQCC result is also around 1 pm, and so the differences in bond length correlate rather well. An analogous result holds for the vibrational frequencies. The MR-AQCC results are consistently about 80 cm

^{-1}lower than the RSS-EOM-CCSD results, which appears rather much. The difference between MR-AQCC and CCSD(T) in the triplet state amounts to 35 cm

^{-1}, the difference due to the dropping of the core is about 15 cm

^{-1}, while on the other hand the RSS-EOM-CCSD results overshoot the CCSD(T) result by about 30 cm

^{-1}. These deviations all point in the same direction and lead to the rather large total deviation of about 80 cm

^{-1}. It is satisfying that this deviation is quite constant between the RSS-EOM-CCSD and the MR-AQCC approaches. The deviation appears slightly less systematic when comparing to the CSS-EOM-CCSD approach.

#### III b. Results for F_{2}.

_{u}LUMO orbital. In this case we cannot compare to experimental results as all of these excited states are dissociative, while also the basis set is not particularly suitable for excited state calculations. Some of the triplet excitation energies can be obtained from ΔCCSDT calculations and they can serve as a benchmark. All calculations have been performed at the CCSD(T)/cc-pVTZ ground state geometry (R=141.36 pm). The singlet and triplet states of a given symmetry have been obtained from state averaged calculations, while in addition we always included complete multiplets in the state-averaged calculations in case of spatial degeneracies (Π and Δ states).

_{2}

^{2-}which is isoelectronic with Ne

_{2}. States that do not contain a deletion involving the σ

_{u}LUMO are in fact doubly excited with respect to the ground state (i.e.

^{1}Δ

_{g},

^{3}Δ

_{u}, ${}_{}^{1}{\sum}_{u}^{+}$, ${}_{}^{3}{\sum}_{u}^{+}$, ${}_{}^{3}{\sum}_{g}^{-}$). Most states (including the ground state) have a fair amount of double excitation character. The singly excited states can be calculated by conventional EE-EOM-CCSD and EE-STEOM-CCSD methods, while the DIP and SS based methods can describe all valence excited states. From tables V (singlet states) and table VI (triplet states) it is seen that quite consistent results are obtained. The agreement between RSS-EOM-CCSD and CSS-EOM-CCSD is good, typically better than 0.1 eV. Moreover for the single determinantal excited triplet states (

^{3}Π

_{u},

^{3}Π

_{g}, ${}_{}^{3}{\sum}_{u}^{+}$) we can compare to ΔCCSD(T) results and also here the comparison with SS-EOMCC is excellent. Since there is little reason to expect some states to be significantly more accurate than others in the SS-EOMCC methods we think that the accuracy is probably good across the board, although we do not have benchmark results to substantiate this assertion. The results from the DIP-STEOM approach are consistently low compared to the other schemes. This might reflect implicit triple excitation effects that are only present in DIP-STEOM, but given the rather poor comparison to CCSD(T) (which includes triple effects explicitly), it appears that the ground state is perhaps somewhat high in energy compared to the excited states in DIP-STEOM. Somewhat surprisingly the EE-EOMCC and EE-STEOMCC methods provide results that can be in significant error (from about 0.2 eV up to 0.5 eV for the ${}_{}^{3}{\sum}_{u}^{+}$ state). This is presumably related to the fact that the ground state of F

_{2}is fairly highly correlated (largest t

_{2}amplitude is -0.18): large electron correlation effects in the parent state in general tend to deteriorate results in EOMCC and STEOMCC calculations. The mediocre results from the not-state-selective EOMCC and STEOM calculations indicate the difficulty to obtain a balanced description of ground and excited states of the fluorine molecule, and provide some perspective as to the high quality of the SS-EOM results. Finally let us discuss if the t coefficients are transferable from one state to the other in an SS-EOMCC calculation. Results are presented in the last three rows of tables V and VI. We have taken the t coefficients of either the ${}_{}^{3}{\sum}_{u}^{+}$ state or the ground state t-coefficients and diagonalized the corresponding transformed Hamiltonian. Very clearly the results are not good at all in this case (errors of 0.5 to 1.5 eV occur). This is a little surprising as these kind of effects do not seem to occur in regular equation of motion coupled cluster calculations or in spin flip EOM as investigated recently by Krylov [26]. One possible explanation is that some correlation effects may be missing in the transfer scheme. For example in the calculations based on the ${}_{}^{3}{\sum}_{u}^{+}$ reference state there are no double excitations that involve the σ

_{g}orbital as it is half empty, while such excitations should be present in the Π and Δ states, but using the cluster amplitudes based on ${}_{}^{3}{\sum}_{u}^{+}$ they are not. Another factor may be that underlying the standard EOM schemes is a many-body similarity transform. Hence the contributions in the final diagonalization from more highly (and neglected) excited determinants can rather easily be analyzed (e.g. [31,49]) through the couplings present in the transformed Hamiltonian. In the current state-selective scheme this strict many-body analysis is discarded and this may well be the source for the lack of transferability of dynamical correlation effects. I regard this aspect a little suspect and hope to investigate the issue more deeply in the future.

_{2}and the singlet states for F

_{2}, (the data for the triplet states of F

_{2}is similar due to the use of the state-averaging strategy). The energy is partioned according to the vacuum determinantal energy corresponding to the formal dianion, the closed shell part of the correlation energy (in a diagrammatic sense) , and the total energy. For comparison in table VII we listed results for Brueckner based CSS-EOM-CCSD and RSS-EOM-CCSD evaluated at identical geometries. First of all we note that even though the formal reference determinant is the same for all states, there are significant differences in vacuum energy that are due to orbital relaxation effects. So even though all states in dioxygen have the same spatial configuration, this does not mean that the optimum spatial orbitals are the same for all states. The ground states in both O

_{2}and F

_{2}have the highest vacuum state energy (corresponding to the dianion), and this reflects the relative compactness of the ground state LUMO. The closed-shell parts of the correlation energy tend to be fairly similar, a clear exception being the closed-shell part of the correlation energy of the ground state for F

_{2}. There are significant differences however between the CSS and RSS variants, in particular the difference in vacuum state energy corresponding to the respective Brueckner determinants. The RSS-EOM-CCSD method universally yields lower total energies than CSS-EOM-CCSD, on the order of 10 mHartree (0.3 eV). The relative energies between various excited states are far more accurate than 0.3 eV, but it does remain a little puzzling where the discrepancy in total energy comes from. Part of it seems to be related to the vacuum energy part which can easily be different by 25 mHartree. It may be good to reiterate that in the RSS approach the reference state $\widehat{R}|{\Phi}_{0}\rangle $ is obtained from a DIP-TDA calculation, diagonalizing the bare Hamiltonian over the 2-hole configurations, while subsequently the t-amplitudes are solved and the vacuum determinant is defined in a Brueckner sense. In contrast in the CSS-EOMCC approach at each step the DIP-EOMCC equations are solved and the reference state $\widehat{R}|{\Phi}_{0}\rangle $ is defined as the 2-hole component of the full DIP-EOM eigenvector. So in the CSS-EOMCC approach the reference state is fully relaxed with respect to dynamical correlation. It is counterintuitive that the RSS-EOMCC approach would yield the lowest energy and this is certainly a clear manifestation of the lack of a variational principle associated with these approaches. The comparison also makes clear that to obtain accurate energy differences we are relying on a balance that leads to a cancellation of errors. I have little insight in the systematics of this balancing act at present and would certainly have preferred to get more consistent total energies.

## IV. Conclusions

_{2}amplitudes are obtained by projecting the parameterized Schrödinger equation $\widehat{\overline{H}}({\widehat{C}}_{1}+{\widehat{C}}_{2})|{\Phi}_{0}\rangle $ onto a suitable excitation manifold, while in the Reference SS-EOMCC scheme we use only the dominant 2-hole part of state and project $\widehat{\overline{H}}\widehat{R}|{\Phi}_{0}\rangle $, where $\widehat{R}$ is obtained from a DIP-TDA calculation. The latter scheme is quite a bit simpler and it provides equally good, possibly better results than the complete variant, in particular for the absolute total energy. Both methods overall perform quite satisfactorily for the diatomic molecules considered, especially for energy differences, and it is premature to make a definitive assessment or comparison between the two approaches. More results and benchmarks need to be obtained. However, if the difference between the two approaches is consistently minor the RCC-EOMCC method is definitely to be preferred, because of its relative simplicity. Let us emphasize here that at present we do not have a rigorous formal argument to choose one approach over the other, as the $\widehat{T}$ amplitudes are abitrary in the limit of a complete expansion. We also investigated if the t-amplitudes that are obtained in this scheme are transferable from one state to the next, and this appears not to be the case. This is a slightly surprising result as the t-amplitudes are supposed to account purely for dynamical correlation, which is expected to be fairly similar between related states. The observations made in this study may guide us in further explorations in our quest for completely general multireference CC methods.

## Acknowledgements

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**Table I.**Internuclear distances (in pm) and harmonic vibrational frequenties in (cm

^{-1}) for the O

_{2}triplet ground state and low-lying singlet states. Calculations in cc-pVTZ basis set.

Method [orbitals] | X${}_{}^{3}{\sum}_{g}^{-}$ | a^{1}Δ_{g} | b${}_{}^{1}{\sum}_{g}^{+}$ | |||

r_{e} | ω_{e} | r_{e} | ω_{e} | r_{e} | ω_{e} | |

Exp.^{a} | 120.74 | 1580.4 | 121.55 | 1509.3 | 122.68 | 1432.7 |

CCSD [ROHF] | 119.63 | 1687 | ||||

CCSD(T) [ROHF] | 120.85 | 1597 | ||||

DIP-EOM-CCSD [2-] | 120.18 | 1634 | 121.05 | 1560 | 122.14 | 1487 |

DIP-STEOM-CCSD [2-] | 120.67 | 1606 | 121.70 | 1508 | 123.10 | 1406 |

RSS-EOM-CCSD [B] | 120.48 | 1629 | 121.38 | 1552 | 122.51 | 1476 |

CSS-EOM-CCSD [B] | 119.94 | 1661 | 121.03 | 1567 | 122.29 | 1481 |

t-RSS-EOM-CCSD [ROHF,X${}_{}^{3}{\sum}_{g}^{-}$] | 120.51 | 1626 | 121.37 | 1552 | 122.56 | 1471 |

t-CSS-EOM-CCSD [ROHF, X${}_{}^{3}{\sum}_{g}^{-}$] | 119.98 | 1658 | 120.60 | 1601 | 121.60 | 1528 |

MRBW-CCSD^{b} | 119.9 | 1683 | 120.4 | 1628 | 121.1 | 1576 |

dropped core MR-AQCC^{c} | 121.72 | 1553 | 122.65 | 1474 | 123.92 | 1391 |

dropped core CCSD(T) [ROHF] | 121.21 | 1585 | ||||

dropped core CCSDT [ROHF] | 121.13 | 1592 |

Method [orbitals] | v_{0,0}(X${}_{}^{3}{\sum}_{g}^{-}$→a^{1}Δ_{g}) | v_{0,0}(X${}_{}^{3}{\sum}_{g}^{-}$→b${}_{}^{1}{\sum}_{g}^{+}$) |

eV | eV | |

Exp.^{a} | 0.9774 | 1.6269 |

DIP-EOM-CCSD [2-] | 0.940 | 1.569 |

DIP-STEOM-CCSD [2-] | 1.014 | 1.700 |

MRBW-CCSD^{b} | 1.058 | 1.902 |

MR-AQCC^{c} | 0.996 | 1.646 |

RSS-EOM-CCSD [B] | 1.040 | 1.708 |

CSS-EOM-CCSD [B] | 0.974 | 1.611 |

t-RSS-EOM-CCSD [ROHF, X${}_{}^{3}{\sum}_{g}^{-}$] | 1.021 | 1.706 |

t-CSS-EOM-CCSD [ROHF, X${}_{}^{3}{\sum}_{g}^{-}$] | 0.995 | 1.696 |

Method [orbitals] | r_{e} (pm) | ω_{e} | Total energy+199 (a.u.) |

Exp.^{a} | 143.5 | 802 | -- |

CCSD [RHF] | 139.23 | 1015 | -.30245 |

CCSD(T) [RHF] | 141.36 | 922 | -.32052 |

DIP-STEOM-CCSD [2-] | 143.53 | 834 | -.30220 |

DIP-EOM-CCSD [2-] | 142.49 | 811 | -.28938 |

CSS-EOM-CCSD [B] | 141.73 | 901 | -.31154 |

RSS-EOM-CCSD [B] | 141.26 | 927 | -.31840 |

**Table IV.**Dominant configurations for ground and excited states of F

_{2}molecule at R=141.36 pm, obtained from DIP-STEOM calculations in cc-pVTZ basis.

Singlet | DIP-Configuration | Triplet | DIP-Configuration | ||

States | / composition | states | / composition | ||

X${}_{}^{1}{\sum}_{g}^{+}$ | ${\sigma}_{u}^{-2}$ | 94.9% | ^{3}Π_{u} | ${\sigma}_{u}^{-1}{\pi}_{g}^{-1}$ | 95.1% |

${\sigma}_{g}^{-2}$ | 4.5% | ${\sigma}_{g}^{-1}{\pi}_{u}^{-1}$ | 4.9% | ||

^{1}Π_{u} | ${\sigma}_{u}^{-1}{\pi}_{g}^{-1}$ | 91.7% | ^{3}Π_{g} | ${\sigma}_{u}^{-1}{\pi}_{u}^{-1}$ | 86.5% |

${\sigma}_{g}^{-1}{\pi}_{u}^{-1}$ | 8.0% | ${\sigma}_{g}^{-1}{\pi}_{g}^{-1}$ | 13.5% | ||

^{1}Π_{g} | ${\sigma}_{u}^{-1}{\pi}_{u}^{-1}$ | 80.1% | ${}_{}^{3}{\sum}_{u}^{+}$ | ${\sigma}_{u}^{-1}{\pi}_{g}^{-1}$ | 99.6% |

${\sigma}_{g}^{-1}{\pi}_{g}^{-1}$ | 19.4% | ||||

^{1}Δ_{g} | ${\pi}_{g}^{-1}{\pi}_{g}^{-1}$ | 2 x 39.3% | ${}_{}^{3}{\sum}_{g}^{-}$ | ${\pi}_{g}^{-1}{\pi}_{g}^{-1}$ | 85.6% |

${\pi}_{u}^{-1}{\pi}_{u}^{-1}$ | 2 x 10.7% | ${\pi}_{u}^{-1}{\pi}_{u}^{-1}$ | 14.4% | ||

${}_{}^{1}{\sum}_{u}^{+}$ | ${\pi}_{u}^{-1}{\pi}_{g}^{-1}$ | 2 x 50.5% | ^{3}Δ_{u} | ${\pi}_{u}^{-1}{\pi}_{g}^{-1}$ | 2 x 50.5% |

**Table V.**Ground state energy (in a.u.) and singlet excitation energies (in eV) for F

_{2}molecule (cc-pVTZ basis, at R=141.36 pm.)

Method [orbitals] | X${}_{}^{1}{\sum}_{g}^{+}$ | ^{1}Π_{u} | ^{1}Π_{g} | ^{1}Δ_{g} | ${}_{}^{1}{\sum}_{u}^{+}$ |

E+199 /au | |||||

CCSD(T) | -.32052 | ||||

EE-EOM-CCSD [RHF] | -.30217 | 4.69 | 7.84 | -- | -- |

EE-STEOM-CCSD [RHF] | -.30217 | 4.50 | 7.73 | -- | -- |

extended-EE-STEOM-CCSD [RHF] | -.30217 | 4.09 | 7.12 | 8.90 | 10.09 |

DIP-EOM-CCSD [2-] | -.30173 | 4.49 | 7.15 | 8.65 | 9.73 |

DIP-STEOM-CCSD [2-] | -.28932 | 4.56 | 7.34 | 8.81 | 9.97 |

RSS-EOM-CCSD [B] | -.31840 | 4.59 | 7.44 | 8.99 | 10.21 |

CSS-EOM-CCSD [B] | -.31153 | 4.71 | 7.50 | 8.95 | 10.27 |

CSS-EOM-CCSD [MCSCF] | -.31068 | 4.68 | 7.46 | 8.91 | 10.13 |

t-RSS-EOM-CCSD [B,${}_{}^{3}{\sum}_{u}^{+}$] | -.31248 | 5.04 | 7.89 | 10.10 | 11.33 |

t-CSS-EOM-CCSD [B, ${}_{}^{3}{\sum}_{u}^{+}$] | -.30913 | 5.06 | 7.91 | 10.14 | 11.38 |

t-RSS-EOM-CCSD [B,X${}_{}^{1}{\sum}_{g}^{+}$] | -.31840 | 5.14 | 8.01 | 10.32 | 11.54 |

Method [orbitals] | ^{3}Π_{u} | ^{3}Π_{g} | ${}_{}^{3}{\sum}_{u}^{+}$ | ${}_{}^{3}{\sum}_{g}^{-}$ | ^{3}Δ_{u} |

ΔCCSD (UHF) | 3.41 | 6.79 | 7.17 | -- | -- |

ΔCCSD(T) (UHF) | 3.44 | 7.01 | 7.07 | -- | -- |

EE-STEOM [RHF] | 3.36 | 7.28 | 6.52 | -- | -- |

DIP-EOM-CCSD [2-] | 3.49 | 6.89 | 7.08 | 8.32 | 10.07 |

DIP-STEOM-CCSD [2-] | 3.29 | 6.63 | 6.73 | 8.13 | 9.84 |

RSS-EOM-CCSD [B] | 3.42 | 6.98 | 7.14 | 8.43 | 10.31 |

CSS-EOM-CCSD [B] | 3.54 | 6.98 | 7.03 | 8.39 | 10.27 |

CSS-EOM-CCSD [MCSCF] | 3.50 | 6.94 | 6.99 | 8.35 | 10.23 |

t-RSS-EOM-CCSD [B, ${}_{}^{3}{\sum}_{u}^{+}$] | 3.87 | 7.36 | 6.98 | 9.55 | 11.43 |

t-CSS-EOM-CCSD [B, ${}_{}^{3}{\sum}_{u}^{+}$] | 3.89 | 7.39 | 7.00 | 9.59 | 11.48 |

t-RSS-EOM-CCSD [B,X${}_{}^{1}{\sum}_{g}^{+}$] | 3.94 | 7.46 | 7.23 | 9.78 | 11.64 |

**Table VII.**Partitioning of various energy terms and total energies for the O

_{2}molecule at 121 pm. and the F

_{2}molecule at R=141.36 pm in the cc-pVTZ basis set.

State | Method | Vacuum energy | Closed part | Total energy |

+ 148 (a.u.) | (a.u.) | +148 (a.u.) | ||

O_{2}, X${}_{}^{3}{\sum}_{g}^{-}$ | CSS-EOM | -0.87206 | -0.50437 | -1.13879 |

RSS-EOM | -0.86963 | -0.52116 | -1.15114 | |

O_{2}, a^{1}Δ_{g} | CSS-EOM | -0.88585 | -0.50731 | -1.10294 |

RSS-EOM | -0.88168 | -0.51925 | -1.11276 | |

O_{2}, b${}_{}^{1}{\sum}_{g}^{+}$ | CSS-EOM | -0.89451 | -0.50729 | -1.07918 |

RSS-EOM | -0.88974 | -0.51973 | -1.08780 | |

+197 (a.u.) | (a.u.) | +197 (a.u.) | ||

F_{2}, X${}_{}^{1}{\sum}_{g}^{+}$ | CSS-EOM | -0.79771 | -.44883 | -2.31068 |

RSS-EOM | -0.69562 | -.51617 | -2.31840 | |

F_{2}, ^{1}Π_{u} | CSS-EOM | -0.84379 | -.55909 | -2.13884 |

RSS-EOM | -0.86501 | -0.56746 | -2.14972 | |

F_{2}, ^{1}Π_{g} | CSS-EOM | -0.85646 | -.56387 | -2.03656 |

RSS-EOM | -0.87738 | -0.56635 | -2.04502 | |

F_{2}, ^{1}Δ_{g} | CSS-EOM | -0.88265 | -.57056 | -1.97068 |

RSS-EOM | -0.90129 | -0.56778 | -1.98809 | |

F_{2}, ${}_{}^{1}{\sum}_{u}^{+}$ | CSS-EOM | -0.88265 | -.57056 | -1.93845 |

RSS-EOM | -0.90129 | -0.56778 | -1.94328 |

© 2002 by MDPI (http://www.mdpi.org).

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Nooijen, M.
State Selective Equation of Motion Coupled Cluster Theory: Some Preliminary Results. *Int. J. Mol. Sci.* **2002**, *3*, 656-675.
https://doi.org/10.3390/i3060656

**AMA Style**

Nooijen M.
State Selective Equation of Motion Coupled Cluster Theory: Some Preliminary Results. *International Journal of Molecular Sciences*. 2002; 3(6):656-675.
https://doi.org/10.3390/i3060656

**Chicago/Turabian Style**

Nooijen, Marcel.
2002. "State Selective Equation of Motion Coupled Cluster Theory: Some Preliminary Results" *International Journal of Molecular Sciences* 3, no. 6: 656-675.
https://doi.org/10.3390/i3060656