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Potential Energy Curves for the Low Lying Electronic States of K+2 from Ab Initio Calculations with All Electrons Correlated Patrycja Skupin, Monika Musial, and Stanislaw Adam Kucharski J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09920 • Publication Date (Web): 27 Jan 2017 Downloaded from http://pubs.acs.org on January 29, 2017

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The Journal of Physical Chemistry

Potential energy curves for the low lying electronic states of K+ 2 from ab initio calculations with all electrons correlated Patrycja Skupin, Monika Musial∗, Stanislaw A. Kucharski Institute of Chemistry University of Silesia Szkolna 9, 40-006 Katowice, Poland January 26, 2017

Abstract The electron affinity (EA) calculations based on the equation-of-motion coupled cluster method proved to be an efficient scheme in the treatment of potential energy curves (PECs) for alkali molecular ions, Me+ 2 . The EA approach provides description of states obtained by an +2 attachment of one electron to the reference which for the Me+ 2 is a doubly ionized Me2 system.

The latter has a very concrete advantage in the calculations of the PECs since it dissociates into + + the closed-shell fragments (Me+2 2 → Me +Me ) hence the restricted Hartree-Fock reference

can be used in the whole range of interatomic distances. In this work accurate potential energy curves and spectroscopic constants are obtained for the six lowest lying electronic states of the K+ 2 ion. The relativistic effects are included by adding appropriate terms of the DK (DouglasKroll) Hamiltonian to the one-electron integrals. ∗

Electronic mail: [email protected]

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Introduction There is a growing interest in the experimental and theoretical studies of the alkali metal di-

atomic molecular systems. The motivation for this is connected with the possible use of alkali diatomics in the studies of ultracold collision and reactions dynamics and other effects occurring in the photodissociation processes.1, 2 The possible applications can be related both to the neutral hetero and homonuclear diatomics as well as to the molecular ions. Hence the accurate quantum chemical calculations can be of help in proper understanding of the nature of the interatomic interactions. However, a majority of theoretical studies focused on the alkali metal diatomics have been based on the model potential methods (MPM) or on the effective core potential (ECP) theories used to replace the effect of the inner-shell electrons. Within this approach the molecular ion Me+ 2 is treated as a system with one electron moving in the field of two atomic cores. This is the case 3–12 also for the K+ are based on the ECP or MPM 2 system for which nearly all theoretical works

approaches with one exception of Ref.3 using open-shell formulation of the coupled cluster (CC) theory. Recently13–21 we have studied dissociation process via the CC theory22–30 for the alkali metal diatomics. The correct description of the potential energy curves (PECs) for the single bond is still a challenge for quantum chemistry methods29, 31 since the homolytic dissociation of a single bond results in the disintegration of the closed-shell molecule into open-shell fragments. The quantum chemical description of such process is difficult since neither RHF (Restricted Hartree Fock) nor UHF (Unrestricted HF) provides a convenient reference function for the whole range of interatomic distances. The ideal situation occurs when the closed-shell molecule dissociates into closed-shell fragments. Such a case arises when we dissociate a double positive ion instead of the neutral molecule or its


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cation in case of alkali dimers. Let us consider as an example the K+ 2 ion which dissociates into the open-shell atom, K, and a closed-shell cation, K+ : K2+ → K . + K + ; K + + K .

(1)

However, the double positive ion of the K+2 2 dissociates as: K2+2 → K + + K +

(2)

and we obtain closed-shell products, K+ , isoelectronic with the Ar atom. The above scheme can be used in the production of PECs for the K+ 2 ion on condition that the applied method is capable to describe the system after attachment of one electron to the reference. In the framework of the coupled cluster theory we have so called EA (electron attachment) scheme formulated within the EOM (equation-of-motion) theory.32–49 In the current study we focus on calculations of the six lowest lying electronic states of K+ 2 ion for large basis sets and EA-EOM-CCSD (S-Singles, D-Doubles) scheme. We use both relativistic approach based on the second-order Douglas-Kroll formalism (DK2)50–55 as well as the standard nonrelativistic formulation. These are the first of such type calculations for this cation with all electrons correlated - on one hand, and also with inclusion of the relativistic corrections - on the other. In the next sections we present the synopsis of the theory and later on we discuss characteristics 2 + 2 + 2 + 2 2 of the six lowest lying electronic states (12 Σ+ g , 1 Σu , 2 Σg , 2 Σu , 1 Πg , 1 Πu ), i.e., PECs together

with selected spectroscopic parameters.


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Methods In the coupled cluster22, 25–30 method the wave function is expressed as: Ψ0 = eT |Φ0 i

(3)

where T is a cluster operator being a sum of operators responsible for the electronic excitations. For the CCSD variant the T operator is represented as T = T1 + T2 . The |Φ0 i is a reference function, i.e., a Slater determinant constructed - in the current approach - with the RHF orbitals. The first step in the calculations is solving the RHF equations and then those of the CCSD method for the reference system: in our case - K+2 2 . The electron attached states, |Ψk i, are parameterized in a linear fashion as: |Ψk i = R(k)|Ψ0 i

(4)

with R(k) = R1 (k) + R2 (k) + · · · =

X a

r a (k)a† +

1 X X ab ri (k)a† b† i + · · · 2 i

(5) (6)

ab

where the standard convention for indices is used, i.e. indices a, b, ... (i, j, ...) refer to virtual (occupied) one-particle levels. The two states in question,Ψ0 and Ψk , satisfy the Schrödinger equations:

and

H|Ψ0 i = E0 |Ψ0 i

(7)

H|Ψk i = Ek |Ψk i

(8)

By choosing to represent the single-electron attached state eigenfunction as in Eq. (4) it is readily obtained that ¯ HR(k) = ωk R(k) ACS Paragon Plus Environment

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(9)

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¯ is the similarity where ωk is the energy change connected with the electron attachment process; H transformed Hamiltonian, in terms of connected diagrams defined as: ¯ = e−T HeT = (HeT )c H

(10)

¯ in the subspace of the determinants conThus, the EA states can be studied by diagonalizing H taining N + 1 electrons (N refers to the number of electrons in |Ψ0 i). The main part of the results is obtained with the EA-CCSD scheme but in one example we are applying also the more advanced model EA-CCSDT (T=Triples) including in full manner the T3 cluster responsible for triple excitations. In our case EA-CCSDT means solving the CC equation for the reference state at the CCSDT level and next solving the EA-EOM equations assuming that the R(k) operator includes also the R3 operator R3 =

1 X X abc r (k)a† b† c† ji 12 abc ij ij

(11)

The size of the EOM matrix, NEOM , can be expressed as NEOM = N1 + N2 + N3 + · · ·

(12)

where N1 ,N2 ,N3 ,. . . is a number of electron-attached determinants generated by the R1 , R2 , R3 , . . ., operators, respectively. For the RHF case the first three NK values are the following N1 = nv

(13)

N2 = no nv (nv − 1)/2 + no n2v

(14)

N3 = no (no − 1)nv (nv − 1)(nv − 2)/12 + n2o n2v (nv − 1)/2 + no (no − 1)n2v (nv − 1)/4

(15)

where no (nv ) is a number of occupied (virtual) levels. The above expressions refer to the number of determinants arising in the spin explicit form of the EOM matrix including doublets, quartets and sextets. All of them refer to the spin z-component Sz = 21 . ACS Paragon Plus Environment

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By introducing spin-adapted form the size of the EOM matrix will be significantly reduced and we may estimate the number of spin-adapted configurations obtained by the Rk operators. It is obvious that R1 creates only the doublet configurations hence we have the following relation ND(R1 ) = N1

(16)

where ND(R1 ) represents the number of doublets created by the R1 operator. Similarly, N2 represents the number of doublets and quartets created by the R2 , and, analogously, N3 includes doublets, quartets and sextets obtained by the action of the R3 operator. The number of quartets included in N2 can be easily calculated as NQ(R2 ) = no nv (nv − 1)/2

(17)

from which the number of doublets at the R2 level follows immediately as ND(R2 ) = N2 − NQ(R2 )

(18)

In analogous manner we may determine the number of spin-adapted configurations generated by the R3 operator: first the sextets NS(R3 ) as NS(R3 ) = no (no − 1)nv (nv − 1)(nv − 2)/12

(19)

next the quartets NQ(R3 ) as NQ(R3 ) = n2o nv (nv − 1)(nv − 2)/6 + no (no − 1)n2v (nv − 1)/4 − NS(R3 )

(20)

and finally the doublets ND(R3 ) as ND(R3 ) = N3 − NQ(R3 ) − NS(R3 )

(21)

In our approach, however, we use generalized Davidson diagonalization scheme to extract only several lowest eigenvalues out of the full spectrum. A crucial step in this procedure is taking product ACS Paragon Plus Environment

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¯ i.e., x = HR. ¯ of the amplitude vector R and the matrix H, So, the schematic EA-EOM-CCSDT equations look like: ¯ EA (k))c |Φo i xa (k) = hΦa |(HR

(22)

ab ¯ EA xab (k))c |Φo i i (k) = hΦi |(HR

(23)

abc ¯ EA xabc (k))c |Φo i ij (k) = hΦij |(HR

(24)

¯ is defined by Eq. (10). This is a standard form of the EOM equations. In practice we where H ¯ matrix but instead we calculate the x vector directly by contraction of avoid construction of the H ¯ elements with R vector. Relevant equations both in the algebraic and diagrammatic form the H were presented in Ref.43

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Results and Discussion 56 A suitable software to carry out calculations for the K+ 2 system seems to be the GAMESS

program package which offers the possibility to do relativistic57 as well as high-level correlated calculations.46, 58, 59 The original version of GAMESS has been supplemented with the local version of the EA-EOM-CC module.37, 43 For all calculations the RHF function for the K+2 cation has 2 been adopted as a reference in accordance with the overall computational strategy presented in previous sections. In principle two basis sets were used in the calculations. The first one denoted as ANO-RCC+ represents uncontracted ANO-RCC basis set of Roos et al.60 with additional two diffuse functions for the s, d shells with exponents 0.00287, 0.00115, 0.10705, 0.04250, respectively. The ANO-RCC+ basis set was used in the standard nonrelativistic calculations. The other basis set employed in second-order Douglas-Kroll relativistic calculations is referred to as SPKRAQZP and is known in the literature as Sapporo-DK-QZP+diffuse set.61 The size of the ANO-RCC+ basis set is 268 and the SPKRAQZP - 196. ACS Paragon Plus Environment

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Atomic calculations In Table 1 we compare the energy of the electronic states of molecular ion at the infinite separation of atoms obtained with the EA-EOM-CCSD to the sum of the energies of the separated K and K+ . For size-extensive methods the two energy values should be identically equal and this situation occurs in the current case. For the ground state we compare the sum of the total electronic energies obtained withe the CCSD scheme for the K+ ion and with the EA-CCSD method for the neutral K atom (see column 4 in Table 1) to the molecular energy for the K+ 2 obtained with EA-CCSD at the interatomic distance R=1000 ˚ A (column 5 in Table 1). For the excited atomic state the energy of the 2 P (4p) and 2 S(5s) terms is compared to the excitation energy of the molecular ion K+ 2 at the dissociation limit. In all cases the compared values are identically the same. The computed values of the energy of 2 P and 2 S terms provide an information on the quality of the basis set. The Douglas-Kroll calculations for the Sapporo basis set provide the energy values differing from experiment by ca. 0.01 eV and the same is true for the ANO-RCC+ basis set used in the nonrelativistic calculations. Molecular calculations We computed potential energy curves for the six states converging to two dissociation limits: K + + K(4s) and K + + K(4p). Due to the fact that the closed-shell reference system (K+2 2 ) dissociates into the closed-shell fragments both SCF and CC calculations easily converge even for large interatomic distances. In Fig. 1 we present the potential energy curves obtained with the Sapporo 2 + + basis set for all considered states. Energies of the 12 Σ+ g and 1 Σu converge to K + K(4s) limit 2 + 2 2 + while remaining four curves 22 Σ+ g , 2 Σu ,1 Πg and 1 Πu converge to the K + K(4p) asymptote.

In Tables 2 and 3 we listed the basic spectroscopic constants like equilibrium geometry, dissociation and excitation energies, harmonic frequencies and anharmonic ωe xe corrections. All these values were obtained with help of LEVEL-8 program of Le Roy.62 For all bound states we evaluated ACS Paragon Plus Environment

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basis set superposition error (BSSE) using counterpoise (CP) correction method. Below we discuss in more detail both sets of PECs. Dissociation limit K + + K(4s) In Fig. 2 two potential energy curves corresponding to the 12 Σ+ g state are shown, the lower one - generated in the relativistic calculations (DK2) and the upper one obtained in the standard nonrelativistic (NR) approach. Both curves represent typical Morse type shape with the minima occurring nearly at the same value of Re . In Table 2 we provide the values of selected spectroscopic constants. The Re obtained at the CCSD level for both types of calculations differs only by 0.004 ˚ A. However, the deviation from the experimental value is larger amounting to ca. 0.11 ˚ A. The De value in DK2 calculations differs from the experiment by 218 wavenumbers while in those based on the NR ANO-RCC+ basis set - by 77 wavenumbers. The computed harmonic frequency differs from the experimental value by 0.1 cm−1 for the DK2 calculations with slightly larger error for the NR results. The anharmonic correction ωe xe is off the experiment by 0.02 cm−1 : 0.18 vs. 0.20 cm−1 both for the DK2 and NR ANO-RCC+ calculations, see Table 2. To assess the quality of our correlated calculations or, in other words, to estimate the importance of the triple excitations, we have done – for the 11 Σ+ g state only – also the EA-EOM-CCSDT calculations. The EA-EOM-CCSDT approach, as described in Ref.,43 relies on the full inclusion of the triple excitation operator both at the ground state level and in the EOM part. In these calculations we used standard ANO-RCC basis set and the results indicate minor effect connected with the full inclusion of the triple excitations. Going from EA-CCSD to EA-CCSDT we observe the change in Re of 0.007 ˚ A and of 7 cm−1 in De , see Table 2. The BSSE error computed at the CCSD level, small but nonnegligible is equal to 5 cm−1 (the CP corrected De value equals to 6588 cm−1 (NR ANO-RCC+ result)).

In the Introduction

we have mentioned that most of the calculations carried out for the alkali metal dimers are based on the model or effective core approximation which reduce the valence electrons problem to solving ACS Paragon Plus Environment

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the one (cation Me+ odinger equation. In order to 2 ) or two-electron (neutral Me2 molecule) Schr¨ assess the effect of inner shells correlation we carried out calculation for the single valence electron +2 calculations for the studied K+ 2 ion with frozen inner electron orbitals taken from the RHF K2

(ANO-RCC+ basis set). The resulting PEC is shown in Fig. 3 (upper curve) and the respective spectroscopic constants are given in Table 2 (see the line next to the experimental values). By freezing the core we introduced relatively large errors. The Re value is off by 0.5 ˚ A and De is too low by ca. 700 cm−1 . We may notice also that the curvature of the of PEC obtained with the frozen core is changed which is revealed in the harmonic frequency value of 63.6 cm−1 (cf. ωe (exp)=73.4 and DK2 result of 73.3 cm−1 ). + In Fig. 4 we demonstrate the DK2 PEC for the 12 Σ+ u state, converging also to the K + K(4s)

limit. The curve shows very shallow minimum with De value of 204 cm−1 obtained in the DK2 calculations (or 82 cm−1 resulting from the NR approach), see Table 2. The computed Re being equal to ca. 11.8 ˚ A indicates a long-range bound state. Although the well depth is small the harmonic frequency value is equal to approximately 6 wavenumbers hence the number of vibrational levels is still large. Note that the BSSE error for this state turns out to be negligible, amounting to 0.02 cm−1 . Dissociation limit K + + K(4p) In the next group of states converging to the K + + K(4p) limit we have one curve corresponding to the 12 Πg state which indicates its repulsive nature, see Fig. 1. Among remaining three curves we 2 2 + have two states, shown in Fig. 5, with well formed minima: 22 Σ+ g and 1 Πu and one state, 2 Σu ,

shown in Fig. 6, for which the bonding effect is very small, of similar nature as the first state of that symmetry (cf. Fig. 4). The energy minimum for the 12 Πu occurs around 5.09 ˚ A (DK2) or 5.05 ˚ A (NR) which indicates a short-bond nature of this state. The DK2 value of De equal to 2847 cm−1 is larger by ca. 400 cm−1 compared to the NR one. A similar relation between DK2 and NR results ACS Paragon Plus Environment

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˚ ˚ is observed for the 22 Σ+ g state. Here the minimum occurs at Re equal to 8.47 A (DK2) and 8.33 A (NR) with with depth of 3207 cm−1 and 2909 cm−1 for the DK2 and NR results, respectively. The excitation energies stay close to each other differing only by 83 cm−1 . ˚ The equilibrium bond length for the 22 Σ+ u state, see Fig. 6, is equal to 15.66 A (DK2) with the De value of 315 cm−1 (DK2, the latter value for the NR case is lower and equal to 182 cm−1 ), see Table 3. The harmonic frequency and the vibrational structure of the 22 Σ+ u state is very close to that of 12 Σ+ u state. −1 The CP correction computed for the 22 Σ+ ). For the other two u state is insignificant (=0.06cm −1 states it amounts to a few wavenumbers: 3 cm−1 for the 22 Σ+ for the 12 Πu one. g state and 8 cm

One can notice that BSSEs are negligible for the states with small dissociation energy since low De value usually occurs for large equilibrium bond length.

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Conclusions The first principle quantum chemical calculations are applied in the theoretical study of the

potential energy curves and spectroscopic constants for the six lowest lying electronic states of the K+ 2 ion. The originality of the results stems from the fact – on one hand – that all 37 electrons were correlated at the EOM-CCSD or EOM-CCSDT level and – on the other – that the relativistic corrections at the DK2 level were taken into account. In order to do that local version of EA-EOMCC module was interfaced to the GAMESS package. The results remain in a satisfactory agreement with available experimental data for the spectroscopic parameters. An important thing in obtaining high accuracy results is an adoption of a proper reference system which in our case is the K+2 2 ion dissociating into a closed-shell fragments which made it possible to use the RHF function for the whole range of interatomic distances. The EAACS Paragon Plus Environment

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EOM-CCSD method being size-extensive (crucial in the PEC calculations) seems to be particularly useful in the study of the dissociation of the alkali molecular ions.


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Acknowledgments This work has been supported by the National Science Centre, Poland under Grant No. 2013/11/B/ST4/02191.


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Paldus, J.; Li, X. Critical Assessment of Coupled Cluster Method in Quantum Chemistry. Adv. Chem. Phys. 1999, 110, 1-175.

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Bartlett, R. J.; Musial, M. Coupled-Cluster theory in Quantum Chemistry. Rev. Mod. Phys. 2007, 79, 291-352.

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Piecuch, P.; Kowalski, K.; Pimienta, I. S. O.; Fan, P. D.; Lodriguito, M.; McGuire, M. J.; Kucharski, S. A.; Ku´s, T.; Musial, M. Method of moments of coupled-cluster equations: A new formalism for designing accurate electronic structure methods for ground and excited states. Theor. Chem. Acc. 2004, 112, 349-393.

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Lyakh, D. I.; Musial, M.; Lotrich, V.; Bartlett, R. J. Multireference nature of chemistry: the coupled-cluster view. Chem. Rev. 2012, 112, 182-243.

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Stanton, J. F.; Bartlett, R. J. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties. J. Chem. Phys. 1993, 98, 7029-7039.

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Krylov, A. I. Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker’s Guide to Fock Space. Annu Rev. Phys. Chem. 2008, 59, 433-462.

34

Nooijen, M.; Bartlett, R. J. Equation of motion coupled cluster method for electron attachment. J. Chem. Phys. 1995, 102, 3629-3647.

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Watts, J. D.; Bartlett, R. J. Economical triple excitation equation-of-motion coupled-cluster methods for excitation energies. Chem. Phys. Lett. 1995, 233, 81-87. ACS Paragon Plus Environment

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Stanton, J. F.; Gauss, J. Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximations. J. Chem. Phys. 1995, 103, 1064-1076.

37

Musial, M.; Kucharski, S. A.; Bartlett, R. J. Approximate inclusion of the T3 and R3 operators in the equation-of-motion coupled cluster method. Adv. Quantum Chem. 2004, 47, 209-222.

38

Kucharski, S. A.; Wloch, M.; Musial, M.; Bartlett, R. J. Coupled-cluster theory for excited electronic states: the full equation-of-motion coupled cluster single, double, and triple excitation method. J. Chem. Phys. 2001, 115, 8263-8266.

39

Kowalski, K.; Piecuch, P. The Active-Space Equation-of-Motion Coupled-Cluster Methods for Excited Electronic States: Full EOMCCSDt. J. Chem. Phys. 2001, 115, 643-651.

40

Hirata, S. Higher-order equation-of-motion coupled-cluster methods. J. Chem. Phys. 2004, 121, 51-59.

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Kamiya, M.; Hirata, S. Higher-order equation-of-motion coupled-cluster methods for electron attachment. J. Chem. Phys. 2007, 126, 134112-1-10.

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Fan P-D.; Kamiya, M.; Hirata, S. Active-space equation-of-motion coupled-cluster methods through quadruple excitations for excited, ionized, and electron-attached states. J. Chem. Theory Comput. 2007, 3, 1036-1046.

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Musial, M.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for electron attached states: EA-EOM-CCSDT. J. Chem. Phys. 2003, 119, 1901-1908.

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Musial, M.; Kucharski, S. A.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT. J. Chem. Phys. 2003, 118, 1128-1136. ACS Paragon Plus Environment

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Musial, M. The excited, ionized and electron attached states within the EOM-CC approach with full inclusion of connected triple excitations. Mol. Phys. 2010, 108, 2921-2931.

46

Gour, J. R.; Piecuch, P. Efficient formulation and computer implementation of the active-space electron-attached and ionized equation-of-motion coupled-cluster methods. J. Chem. Phys. 2006, 125, 234107-234117.

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Gour, J. R.; Piecuch, P.; Wloch, M. Active-space equation-of-motion coupled-cluster methods for excited states of radicals and other open-shell systems: EA-EOMCCSDt and IP-EOMCCSDt. J. Chem. Phys. 2005, 123, 134113-1-14.

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Gour, J. R.; Piecuch, P.; Wloch, Extension of the Active-Space Equation-of-Motion CoupledCluster Methods of Radical Systems: The EA-EOMCCSDt and IP-EOMCCSDt Approaches. International Journal of Quantum Chemistry 2005, 106, 2855-2874.

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Ehara, M.; Piecuch, P.; Lutz, J. J.; Gour, J. R. Symmetry-adapted-cluster configurationinteraction and equation-of-motion coupled-cluster studies of electronically excited states of copper tetrachloride and copper tetrabromide dianions. Chem. Phys. 2012, 399, 94-110.

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Douglas, N.; Kroll, N. M. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 1974, 82, 89-155.

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Hess, B. A. Applicability of the no-pair equation with free-particle projection operators to atomic and molecular structure calculations. Phys. Rev. A 1985, 32, 756-763.

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Hess, B. A. Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Phys. Rev. A 1986, 33, 3742-3748.

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Jensen,G.; Hess, B. A. Revision of the Douglas-Kroll transformation. Phys. Rev. A 1989, 39, 6016-6017. ACS Paragon Plus Environment

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Barysz, M.; Sadlej, A. J. Two-component methods of relativistic quantum chemistry: from the DouglasKroll approximation to the exact two-component formalism. J. Mol. Struct., (THEOCHEM) 2001, 573, 181-200.

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Wolf, A.; Reiher, M.; Hess, B. A. The generalized Douglas-Kroll transformation. J. Chem. Phys. 2002, 117, 9215-9226.

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Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. J.; Dupuis, M.; Montgomery, J. A. General atomic and molecular electronic structure system. J. Comput. Chem. 1993, 14, 1347-1363.

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Fedorov, D.; Schmidt, M. W.; Koseki, S.; Gordon, M. S. In Recent Advances in Relativistic Molecular Theory; Hirao, K., Ishikawa, Y., Eds.; WSP: Singapore, 2004; Vol. 5, pp 107-136.

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Piecuch, P.; Kucharski, S. A.; Kowalski, K.; Musial, M. Efficient computer implementation of the renormalized coupled cluster methods. The R-CCSD[T], R-CCSD(T), CR-CCSD[T] and CR-CCSD(T) approaches. Comp. Phys. Commun. 2002, 149, 71-96.

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Page 21 of 38

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Le Roy, R. J. LEVEL 8.0: A Computer Program for Solving the Radial Schrödinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663 (2007); see http://leroy.uwaterloo.ca/programs.

63

Broyer, M.; Chevaleyre, J.; Ddacretaz, G.; Martin, S.; Wöste, L. K2 rydberg state analysis by two-and three-photon ionization. Chem. Phys. Lett. 1983, 99, 206-212.

64

Leutwyler, S.; Herrmann, A.; Wöste, L.; Schumacher, E. Isotope selective two-step photoionization study of K2 in a supersonic molecular beam. Chem. Phys. 1980, 48, 253-267.

65

Falke, S.; Tiemann, E.; Lisdat, C.; Schantz, H.; Grosche, G. ransition frequencies of the D lines of

39

K,

40

K, and

41

K measured with a femtosecond laser frequency comb. Phys. Rev. A 2006,

74, 032503-1-9.


21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 38

Figure caption:

Figure 1. Potential energy curves for the six lowest lying electronic states of K+ 2 with the EA-EOMCCSD DK2 method and SPKRAQZP basis set.

+ Figure 2. Potential energy curves for the 12 Σ+ g state of K2 with the EA-EOM-CCSD method and

ANO-RCC+ (NR) and SPKRAQZP (DK2) basis sets.

+ Figure 3. Potential energy curves for the 12 Σ+ g state of K2 with the regular EA-EOM-CCSD/ANO-

RCC+ (NR) method and frozen core 1 electron/ANO-RCC+ approximation (NR).

+ Figure 4. Potential energy curve for the 12 Σ+ u state of K2 with the EA-EOM-CCSD DK2 method

and SPKRAQZP basis set.

+ 2 Figure 5. Potential energy curves for the 22 Σ+ g and 1 Πu states of K2 with the EA-EOM-CCSD

method and ANO-RCC+ (NR) and SPKRAQZP (DK2) basis sets.

Figure 6. Potential energy curve for the 22 Σu state of K+ 2 with the EA-EOM-CCSD DK2 method and SPKRAQZP basis set.


22

Page 23 of 38

Fig. 1

K2+ (SPKRAQZP/DK2) -1203.260 2

+

1 Σg 2

1 Σu+

-1203.280

2

2 Σg

+

2

2 Σu+

-1203.300

12Πg

-1203.320

2

1 Πu

-1203.340 E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1203.360 4p -1203.380 -1203.400 -1203.420

4s

-1203.440 -1203.460 0.0

5.0

10.0

15.0

20.0

R (Angstrom)


23

25.0

30.0


Fig. 2

12Σ+g ANO−RCC+

−1199.340

12Σ+g SPKRAQZP

−1199.360

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 38

−1199.380 −1203.420

−1203.440

−1203.460 4

6

8

10

12 R (Angstrom)

14


24

16

18

20

Page 25 of 38

Fig. 3

+ K2 (ANO-RCC+)

-1199.320 2

+

X Σg all electrons EA-EOM-CCSD (NR) 2 + X Σg frozen core 1 el. approx. (NR)

-1199.330

-1199.340

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1199.350

-1199.360

-1199.370

-1199.380 0.0

5.0

10.0 R (Angstrom)


25

15.0

20.0


Fig. 4

-1203.423 12Σu+

-1203.424

-1203.424

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 38

-1203.425

-1203.425

-1203.426

-1203.427 8.0

10.0

12.0

14.0

16.0

18.0

R (Angstrom)


26

20.0

22.0

24.0

Page 27 of 38

Fig. 5

−1199.260 22Σg+ ANO−RCC+ 2

1 Πu ANO−RCC+ 22Σg+ SPKRAQZP 2

1 Πu SPKRAQZP −1199.280

−1199.300 E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−1203.360

−1203.380

4

6

8

10

12 14 R (Angstrom)

16


27

18

20

22


Fig. 6

-1203.365 22Σu+

-1203.365

-1203.366

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 38

-1203.366

-1203.367

-1203.367

-1203.368 10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

R (Angstrom)


28

26.0

28.0

30.0

Page 29 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1. K+ 2 ground and excited state energies for separated atoms. Dissociation

K+

K

K+ +K

limit

CCSD

EA-EOM-CCSD

˚ K+ 2 (R = 1000 A)

Exp.a)

EA-EOM-CCSD

ANO-RCC+ (NR) 1

1

S(K + ) +2 S(K)b)

-599.595142

-599.753899

-1199.349041

-1199.349041

-

1.6067

1.6067

1.6067

1.617

2.5917

2.5917

2.5917

2.607

-601.792434

-1203.425536

-1203.425536

-

1.6063

1.6063

1.6063

1.617

2.5949

2.5949

2.5949

2.607

S(K + ) +2 P (K)c)

1

S(K + ) +2 S(K)c)

SPKRAQZP DK2 1

1

S(K + ) +2 S(K)b)

-601.633102

S(K + ) +2 P (K)c)

1

S(K + ) +2 S(K)c) a)

Ref.65

b)

Total ground state (1 S(K + ) +2 S(K)) energy in a.u.

c)

Excitation energies in eV.


29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 38

Table 2. Selected spectroscopic constants of K2+ (dissociation limit: K + + K(4s)). Sym.

Re (˚ A)

De (cm−1 )

Te (cm−1 )

ωe (cm−1 )

ωe xe (cm−1 )

Basis set

Method

12 Σ+ g

4.435

6 481

0

72.4

0.13

ANO-RCC

EA-EOM-CCSD (NR)

4.438

6 474

0

72.2

0.13

ANO-RCC

EA-EOM-CCSDT (NR)

4.519

6 593

0

72.8

0.18

ANO-RCC+

EA-EOM-CCSD (NR)

4.515

6 811

0

73.3

0.18

SPKRAQZP

EA-EOM-CCSD (DK2)

4.895

5964

0

63.6

0.14

ANO-RCC+

f. c. 1 el. app.a) (NR)

4.4

6 670

0

73.4

0.20

11.849

82

6 511

5.7

0.11

ANO-RCC+

EA-EOM-CCSD (NR)

11.748

204

6 608

5.8

0.11

SPKRAQZP

EA-EOM-CCSD (DK2)

12 Σ+ u

a)

exp.63, 64

frozen core 1 electron approximation.


30

Page 31 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3. Selected spectroscopic constants of K2+ (dissociation limit: K + + K(4p)). Sym.

Re (˚ A)

De (cm−1 )

Te (cm−1 )

ωe (cm−1 )

ωe xe (cm−1 )

Basis set

Method

22 Σ+ g

8.325

2 909

16 643

30.2

0.04

ANO-RCC+

EA-EOM-CCSD (NR)

8.466

3 207

16 560

28.6

0.04

SPKRAQZP

EA-EOM-CCSD (DK2)

15.295

182

19 370

6.1

0.06

ANO-RCC+

EA-EOM-CCSD (NR)

15.659

315

19 453

5.7

0.05

SPKRAQZP

EA-EOM-CCSD (DK2)

repulsive

ANO-RCC+

EA-EOM-CCSD (NR)

repulsive

SPKRAQZP

EA-EOM-CCSD (DK2)

22 Σ+ u

12 Πg

12 Πu

5.053

2 463

17 089

40.1

0.09

ANO-RCC+

EA-EOM-CCSD (NR)

5.094

2 847

16 920

39.0

0.13

SPKRAQZP

EA-EOM-CCSD (DK2)


31


K2+ (SPKRAQZP/DK2) -1203.260 12Σg+ 12Σu+

-1203.280

22Σg+ 22Σu+

-1203.300

12Πg

-1203.320

12Πu

-1203.340 E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 32 of 38

-1203.360 4p -1203.380 -1203.400 -1203.420

4s

-1203.440 -1203.460 0.0

5.0

10.0

15.0 R (Angstrom)


20.0

25.0

30.0

Page 33 of 38

12Σ+g ANO−RCC+

−1199.340

12Σ+g SPKRAQZP

−1199.360

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−1199.380 −1203.420

−1203.440

−1203.460 4

6

8

10

12 R (Angstrom)

14

16

18


20


K2+ (ANO-RCC+) -1199.320 X2Σg+ all electrons EA-EOM-CCSD (NR) X2Σg+ frozen core 1 el. approx. (NR) -1199.330

-1199.340

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 34 of 38

-1199.350

-1199.360

-1199.370

-1199.380 0.0

5.0

10.0 R (Angstrom) ACS Paragon Plus Environment

15.0

20.0

Page 35 of 38

-1203.423 12Σu+

-1203.424

-1203.424

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


-1203.425

-1203.425

-1203.426

-1203.427 8.0

10.0

12.0

14.0

16.0 R (Angstrom)


18.0

20.0

22.0

24.0


−1199.260 22Σg+ ANO−RCC+ 12Πu ANO−RCC+ 22Σg+ SPKRAQZP 12Πu SPKRAQZP −1199.280

−1199.300 E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 38

−1203.360

−1203.380

4

6

8

10

12 14 R (Angstrom)


16

18

20

22

Page 37 of 38

-1203.365 22Σu+

-1203.365

-1203.366

E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


-1203.366

-1203.367

-1203.367

-1203.368 10.0

12.0

14.0

16.0

18.0

20.0 22.0 R (Angstrom)


24.0

26.0

28.0

30.0


K2+ (EA-EOM-CCSD DK2/SPKRAQZP) -1203.260 12Σg+ 12Σu+

-1203.280

22Σg+ -1203.300

22Σu+ 12Πg

-1203.320

12Πu

-1203.340 E (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 38 of 38

-1203.360 4p -1203.380 -1203.400 -1203.420

4s

-1203.440 -1203.460 0.0

5.0

10.0

15.0 R (Angstrom)


20.0

25.0

30.0

Potential Energy Curves for the Low-Lying ... - ACS Publications

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