Potential Energy Curves for the Low-Lying Electronic States of KLi with Fock Space Coupled Cluster Method

CHAPTER EIGHT

Potential Energy Curves for the Low-Lying Electronic States of KLi with Fock Space Coupled Cluster Method Monika Musiał*,1, Anna Motyl*, Patrycja Skupin*, Stanisław A. Kucharski* *Institute of Chemistry, University of Silesia, Katowice, Poland 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Synopsis of the Theory 3. Results and Discussion 4. Conclusions Acknowledgments References

201 203 205 211 212 212

Abstract Accurate potential energy curves (PECs) are obtained for 10 lowest lying electronic states of the KLi molecule. Contrary to the previous studies, the current approach is based on the first-principle calculations with all electrons correlated. The computational scheme used here is based on the multireference coupled cluster theory formulated in the (2,0) sector of the Fock space. The latter sector provides description of the states obtained by attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionized KLi2+ structure as a reference. The latter has a very concrete advantage in the calculations of the PECs since they dissociate into closed shell fragments (KLi2 + ! K++Li+); hence, the restricted Hartree–Fock reference can be used in the whole range of interatomic distances.

1. INTRODUCTION The heteronuclear alkali dimers offer an additional feature in the experimental studies due to their permanent dipole moment and the Advances in Quantum Chemistry, Volume 72 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2015.05.001

#

2016 Elsevier Inc. All rights reserved.

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possibility to influence their behavior with the external electric field. The KLi molecule is such an example of the mixed alkali diatomics and due to that it attracts a lot of interest both from the experimentalists1–14 and theoreticians.15–18 The common feature of the KLi papers is a focus on the interatomic potential since its accurate description enables explanation of the molecular behavior in many situations. The standard approach to the calculation of the potential energy curves (PECs) in alkali metal diatomics is based on replacing the inner-shell electrons with the effective or model potential.15–18 Hence, the effect of the core electrons is modeled with adjustable parameters, and actual calculations are limited to two valence electrons. The obvious method used in the latter case is the configuration interaction scheme for the single and double excitations which in this case is a full CI (FCI) treatment. The FCI approach has many advantages, being variational, size extensive, and invariant with respect to the unitary transformation among orbitals. The problem with the adequate treatment of the dissociation process of the closed shell molecule is caused by the fact that the reference function based on the restricted Hartree–Fock (RHF) calculations cannot correctly describe the system at large interatomic distances. In the homolytic dissociation, the single-bonded closed shell molecule AB falls apart into open shell fragments: AB!A. +B. and this requires using the unrestricted HF (UHF) scheme. The latter, however, has certain disadvantages connected with the space and spin symmetry as well as some convergence problems around the critical geometries. The solution to the problem would be a selection of the method based on the RHF reference in the whole range of interatomic distances. For alkali metal diatomics, such a situation arises when the reference function is obtained for the doubly positive molecular ion. In this case, the closed shell ion AB2+ dissociates into closed shell fragments according to the equation AB2 + ! A + + B +

(1)

In our example, the doubly positive ion KLi2+ dissociates into closed shell fragments K+ and Li+. The above approach can be used in the production of PECs for neutral KLi molecule on condition that the applied quantum chemical method is capable to describe the system after attachment of two electrons to the reference, i.e., after recovering the original structure. In the framework of the coupled cluster (CC)19–28 theory, we have so-called DEA (double electron attachment) schemes29–37: one formulated

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within the EOM (equation-of-motion) theory29,37 and the other based on the multireference coupled cluster (MRCC) method formulated in the (2,0) sector of the Fock space (FS).30–36 The latter scheme, however, seems particularly suitable for the bond breaking problem due to its rigorous size extensivity and is used here to produce PECs for the low-lying states of the KLi molecule. Thus, the computational scheme used in this work relies (i) on removal of two electrons from the system to have a possibility of using the RHF reference and (ii) on recovering the original neutral system by using the postHF scheme capable to describe the addition of two electrons to the system.

2. SYNOPSIS OF THE THEORY In this section, we give several basic definitions allowing to introduce the method used in the calculations. The CC19–28 approach is based on the exponential parameterization of the wave function Ψ Ψo ¼ eT Φo

(2)

where T ¼ T1 + T2 is a cluster operator being—in the CCSD model adopted in this work—a sum of operators responsible for the single (T1) and double (T2) excitations and the Φo being the reference function, i.e., a Slater determinant constructed from the RHF orbitals. Thus, the first step in the current calculations is solving the RHF equations and the next one is providing the CCSD solution for the reference system. Note that by the reference system, we understand the doubly ionized structure KLi2+ and both RHF and CCSD solutions are obtained for the latter. The principal idea of the MR approaches38 is to solve the eigenvalue equation for the effective Hamiltonian operator, Heff: Heff Ψo ¼ EΨo

(3) 38–46

defined within a model space. The model space in the FS formalism is the configurational subspace of the FS obtained as a direct sum of the  particular sectors M ði, jÞ ðk, lÞ

M

¼

i¼k , j¼l M



M ði, jÞ

(4)

i, j¼0  The sector M ði, jÞ is a configurational space obtained by the creation of i particles within the mp active virtual orbitals and of j holes within the

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mh active occupied orbitals. The active (called also valence) orbitals form the one-electron valence space composed of two parts: hole subspace and particle subspace. Hence, an unequivocal definition of the model space is indicated by giving the top sector (k,l) and the size of the active particle (mp) and hole (mh) spaces. The double electron attached states correspond to the (2,0) sector and we may write i¼2 M

Mð2, 0Þ ¼



M ði, 0Þ

(5)

i¼0

In this case, the active space includes only particle levels; hence, providing mp number (hereafter denoted by m) is enough to define the active space. In the (2,0) sector of the FS, the model space is spanned by the configurations Φαβ formed by distributing the added two electrons in all possible ways among the m valence virtual levels. The hierarchical nature of the FS solutions requires that in order to solve the FS equations for the (k,l) sector, all the solutions for lower rank sectors (i, j) with i  k and j  l must be known. In the current case, this requirement enforces solving the FS equations for the (0,0) and (1,0) sectors. The (0,0) sector corresponds to the mentioned above single reference solution for the reference system, i.e., KLi2+ ion. The (1,0) sector is spanned by the mΦα configurations formed by placing an additional electron in one of the m valence levels. The energy values of the double electron attached states are obtained by the diagonalization of the effective Hamiltonian within the Φαβ configurational space: ð2, 0Þ

Heff

,

ð0 0Þ

¼ Pð2, 0Þ HeS

+ Sð1, 0Þ + Sð2, 0Þ ð2, 0Þ

where the projection operator P(2,0) is defined as X jΦαβ ihΦαβ j Pð2, 0Þ ¼ αβ

P

(6)

(7)

The S(0,0)( T), S(1,0), and S(2,0) are cluster operators for sector indicated by the superscripts. Although, in order to construct Heff, the amplitudes from the lower rank sectors, i.e., (0,0) and (1,0), are needed, the diagonalization of the Heff is carried out within the (2,0) sector only. The operator S(1,0) can be obtained by solving the respective FS-MRCC (1,0)47–49 equations but the same result can be obtained with the

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EA-EOM-CC (EA—electron affinity) scheme, i.e., the EOM scheme applied to the single electron attached states.50,51 We know that the energy values obtained with the EA-EOM-CC scheme are identical to those of FS-MRCC for (1,0) sector while the corresponding eigenvectors differ only by a straightforward transformation. Once the S amplitudes for the (0,0) and (1,0) sectors are found, we can set up the equations for the S(2,0) amplitudes which subsequently are used to construct the Heff operator, the diagonalization of which provides the sought energies of the double electron attached states. In order to eliminate convergence problems in the (2,0) sector, we applied intermediate Hamiltonian (IH) strategy50–56 described in detail in our paper devoted to the FS-CCSD (2,0) method.30

3. RESULTS AND DISCUSSION All calculations are done using the ACES II57 program system supplemented with the FS-CCSD (2,0) module.30 In the double electron attachment calculations, the orbitals used are obtained by the RHF solution for the KLi2+ system in accordance with the overall computational strategy described in Section 1. We used the POL158 basis set with the spherical harmonic polarization functions and all electrons were correlated. The size of the active space for the FS-CCSD (2,0) part has been set to m ¼ 44 (i.e., 44 lowest virtual orbitals have been selected as active). Note that the resulting size of the model space is equal to 1936, i.e., that many configurations span the subspace in which the effective Hamiltonian is being diagonalized. In Table 1, we quote the energy values of the Li and K atoms. In order to be able to relate the atomic results to the molecular values, we have to use in atomic calculation also the FS-type method. The respective sector here is one valence (1,0), i.e., we study the system obtained upon addition of the electron to the Li+ and K+ ions in full analogy to the mentioned above treatment of the (1,0) sector in the KLi calculations. Note that in this case, the reference function is also RHF type since both ions are closed shell structure isoelectronic with the He and Ar noble gas atoms, respectively. In real calculations, we replace the FS-CCSD (1,0) scheme with the EA-EOMCCSD due to mentioned earlier identity of both approaches and the fact that EOM computations are much easier to carry out. For both atoms, we quote the energies of the three lowest lying excited states. The computed values remain in a good agreement with the

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Table 1 Excitation Energies (eV) for the Li and K Atoms in POL1 Basis Set EA-EOM-CCSD Sym. ( FS-CCSD (1,0)) Δ

Exp.a

Li 2 0

P (2p)

1.839

0.009

1.848

2

S (3s)

3.355

0.018

3.373

3.816

0.0184

3.834

2 0

P (3p)

MAE

b

0.015

K 2 0

P (4p)

1.614

0.007

1.607

2

S (5s)

2.556

0.051

2.607

2

D (3d)

2.727

0.057

2.670

MAEb a

0.038

Ref. 59 for Li; Ref. 60 for K. Mean absolute error.

b

Table 2 Energies of the Electronic States at the Dissociation Limit of the KLi Molecule Compared to the Atomic Energies in the POL1 Basis Set Ka Li+K KLi(R5∞) Lia

Config.

E(a.u.)

Config.

E(a.u.)

E(a.u.)

E(a.u.)

[He]2s

7.421285

[Ar]4s

599.275607

606.696892

606.696892

ΔE(eV)

ΔE(eV)

ΔE(eV)

ΔE(eV) [He]2s

0

[Ar]4p

1.614215

1.614215

1.614215

[He]2p

1.838727

[Ar]4s

0

1.838727

1.838727

[He]2s

0

[Ar]5s

2.555957

2.555957

2.555957

a

Energy calculated using EA-EOM-CCSD( FS-CCSD (1,0)) method.

experiment. For the Li atom, the error stays between 0.01 and 0.02 eV, for the K atom case is of the order of 0.05 eV. The atomic energies are important in the current calculations since the energies of the electronic states of the KLi molecule for size-extensive methods should converge at infinite distance to the atomic values. In the first row of Table 2, we compare the total ground-state energy of the KLi molecule at R ¼ 1 (last column) with the sum of the ground-state energies

207

Potential Energy Curves for the Low-Lying Electronic States

of the Li and K atoms. The results are identical. Note that the ground-state energy is obtained in the same scheme as the excited-state energy values, i.e., within the framework of the method defined in the (2,0) sector of the FS. In the second part of Table 2, we compare the atomic excitation energies with the three lowest energy states of the KLi molecule at infinite interatomic distance. We see that in the supermolecular calculations, the excitation energy of the KLi is identically equal to the atomic excitation energies, shown in Table 1. Note that in all cases, the process of double electron attachment engages both involved centers since the two added electrons are placed each in one atom. This observation is important since it tells us that the size-intensive methods (like DEA-EOM) are not going to give the size-extensive results contrary to the FS-MRCC approach. In the current work, we have studied PECs for the lowest lying electronic states of the KLi molecule. These are the states correlating to the three dissociation limits: Li(2s)+K(4s)—2 states, Li(2s)+K(4p)—4 states, and Li(2p) +K(4s)—4 states. In the cases where the p atomic levels are engaged, some of the molecular states (Π) are degenerated, and hence, the total number of state eigenfunctions amounts to 14. In Fig. 1, we present the PECs for the X1Σ+ state of the KLi molecule, obtained with the multireference FS-CCSD (2,0) scheme (solid line). Since KLi: ground state (POL1 basis set) −606.660 −606.670

E (a.u.)

−606.680 −606.690 −606.700 −606.710 CCSD CCSDT FS-CCSD (2,0)

−606.720 −606.730 0.0

5.0

10.0 R (Å)

15.0

20.0

Figure 1 Potential energy curves for the GS of the KLi molecule with various CC methods in the POL1 basis set.

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this is a ground state, we may compare it with the curves generated with the single reference methods. We selected CCSD and CCSDT, keeping in mind that the latter may be considered here as the reference curve. We see that the FS curve, obtained at the CCSD level, stays very close and is fully parallel to that computed at much higher cost, i.e., with full inclusion of the triple excitations. The single reference CCSD method provides the curve with unreasonably high dissociation limit. In the following three figures, Figs. 2–4, we show the PECs corresponding to the electronic states correlating to the mentioned above three dissociating limits. In Fig. 2, we plot two curves: one representing the discussed above ground X1Σ+ state and the other, the triplet 13Σ+ state. Both have the same asymptotic limits, Li(2s)+K(4s). We see that the curve corresponding to the triplet state shows a small bonding effect and dissociation energy (De) of 376 cm1, see Table 3, which compares well with the experimental value of 287 cm1.5 A very good agreement of the computed De value with the experiment is obtained for the X1Σ+ state, with the error of 9 cm1. Note also a satisfactory agreement between computed and experimental ωe values (error of 2 cm1). The adiabatic excitation energy of the lowest triplet state is off by 98 wavenumbers. The equilibrium geometry for ˚. both states is reproduced with the accuracy of ca. 0.03 A KLi (POL1 basis set) −606.600 −606.620

E (a.u.)

−606.640 −606.660 −606.680 2s+4s

−606.700 X1Σ+

−606.720

3 +

1Σ 1 +

−606.740 0.0

X Σ EXP

5.0

10.0

15.0

20.0

R (Å)

Figure 2 Potential energy curves for the KLi molecule with the FS-CCSD (2,0) method for Li(2s)+K(4s) dissociation limit in the POL1 basis set. Experimental curve from Ref. 4.

209

Potential Energy Curves for the Low-Lying Electronic States

KLi (POL1 basis set) −606.540

1 +

2Σ

3 +

2Σ

−606.560

1

1Π 3

1Π 21Σ+ EXP

E (a.u.)

−606.580

11Π EXP

−606.600 −606.620 2s+4p

−606.640 −606.660 −606.680 0.0

5.0

10.0

15.0

20.0

R (Å)

Figure 3 Potential energy curves for the KLi molecule with the FS-CCSD (2,0) method for Li(2s)+K(4p) dissociation limit in the POL1 basis set. Experimental curves from Refs. 9,10.

KLi (POL1 basis set) −606.540

1 +

3Σ

3 +

3Σ 1

−606.560

2Π 3

2Π 31Σ+ EXP 21Π EXP

E (a.u.)

−606.580

−606.600

−606.620 2p+4s

−606.640

−606.660 0.0

5.0

10.0

15.0

20.0

R (Å)

Figure 4 Potential energy curves for the KLi molecule with the FS-CCSD (2,0) method for Li(2p)+K(4s) dissociation limit in the POL1 basis set. Experimental curves from Refs. 11,12.

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Table 3 Spectroscopic Constants for KLi Molecule (FS-CCSD (2,0) Method, POL1 Basis Set) De(cm21) Te(cm21) we(cm21) wexe(cm21) Sym. Re(Å)

Ref.

Dissociation limit Li(2s)+K(4s)

X 1 Σ+ 13 Σ +

3.296

6208

0

210.02

1.32

3.326

6217

0

212.04

–

4.966

376

5832

40.85

0.94

4.992

287

5930

–

–

Exp.1,3,5 Exp.5

Dissociation limit Li(2s)+K(4p)

21 Σ +

3.949

7202

12,026

132.67

0.77

3.947

7105

12,097

137.08

–

23 Σ +

3.930

4267

14,961

139.87

1.17

11 Π

3.729

1496

17,732

122.15

4.25

3.713

1686

17,573

135.84

–

3.236

8702

10,525

204.17

0.87

13 Π

Exp.9

Exp.10

Dissociation limit Li(2p)+K(4s)

31 Σ +

4.152

3369

17,669

110.92

0.32

4.192

3619

17,501

115.41

–

33 Σ +

3.770

875.63

21,914

163.93

4.03

2Π

4.000

1682

19,356

134.11

1.91

4.043

1664

19,456

128.98

–

4.109

674

20,365

96.49

3.62

0.026

110

119

5.95

–

1

23 Π MAE a

a

Exp.11

Exp.13

Mean absolute error.

The next group of curves is plotted in Fig. 3. These are the four curves correlating to the Li(2s)+K(4p) asymptote corresponding to the states: 21Σ+, 23Σ+, 11Π, and 13Π. All four states are of bonding character with the largest well depth of 8702 cm1 observed for 3Π state (see Table 3). For singlet states, the experimental values are available and given in Ref. 9 for the 1 + Σ state and in Ref. 10 for 1Π state. For the 1Σ+ state, the applied method reproduces very well the available experimental data. The equilibrium bond length is obtained with the

Potential Energy Curves for the Low-Lying Electronic States

211

˚ while the adiabatic excitation energy is off by accuracy of 0.002 A 71 wavenumbers, i.e., below 0.01 eV. The theoretical result of similar quality is obtained also for the dissociation energy, cf. 7202 cm1 versus experimental value of 7105 cm1. A slightly worse agreement between theoretical and experimental values is obtained for the 11Π state. Here, the equilibrium bond length computed ˚ , while the dissociation energy is with FS-CCSD scheme is larger by 0.016 A 1 smaller by 190 cm . The adiabatic excitation energy, Te, computed in the current work is equal to 17,732 cm1 which is off the experimental value10 by 159 cm1. The satisfactory agreement of the computed and experimental potential can be observed in Fig. 3 where for the singlet curves we plotted also the experimental curves. We may see that the curves match each other very well both for the 1Σ+ and 1Π states. The following four states: 31Σ+, 21Π, 23Π, and 33Σ+, dissociate to the Li(2p)+K(4s) limit. The corresponding PECs are shown in Fig. 4. Three of them, 31Σ+, 21Π, and 23Π, are of regular Morse-type shape, while the fourth one, 33Σ+, has a barrier 893 cm1 high occurring around 4.9 A˚. Note that the same barrier is reported in the other theoretical papers, e.g., Ref. 15. For both singlet states, similarly as in the previous group, the experimental data are available. In Table 3, we quote the theoretical and experimental values for the considered spectroscopic constants. The best agreement with the experiment is observed for the adiabatic excitation energy, Te, with the errors equal to 168 cm1 (31Σ+) and 100 cm1 (21Π). The dissociation energy, De, for the 31Σ+ state is off by 250 cm1 and by only 18 cm1 for the 21Π state. The equilibrium geometry for both states is reproduced with the error of ˚ . Note that the 23Π state is very weakly bound with the De equal 0.04 A to 674 cm1.

4. CONCLUSIONS The first-principle method has been applied to the theoretical study of the PECs for 10 low-energy electronic states of the KLi molecule. In spite of the modest basis set (POL1 with total number of basis functions equal to 64), the results are quite satisfactory and reproduce the spectroscopic constants with good accuracy, e.g., the mean absolute error, calculated for these states for which the experimental values are available, is equal to 110 cm1 for the dissociation energy (six states) and 119 cm1 for the adiabatic excitation energy (five states) (see Table 3). We observe also a very good

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agreement of the theoretical PECs with the experimental ones if the latter are available (see Figs. 3 and 4). We emphasize that these were rigorous ab initio calculations with all 22 electrons correlated. We believe that the crucial thing in obtaining smooth and correct PECs from the equilibrium up to infinity is an adoption of the DEA strategy which relies (i) on the removal of a pair of electrons from the KLi molecule to obtain a convenient RHF reference and (ii) on using the DEA scheme to recover the original neutral structure upon attaching two electrons to the KLi2+ ion. The method best suitable for the second step seems to be a rigorously size-extensive MRCC scheme formulated within the (2,0) sector of the FS. We expect that the substantial enlargement of the basis set may improve the results and the spectroscopic accuracy can be achieved upon inclusion of the relativistic effects into calculations.

ACKNOWLEDGMENTS It is a great pleasure to contribute this chapter to a special volume in honor of Professor Frank E. Harris on the occasion of his 85th birthday. One of us (M.M.) would like to express her gratitude for the hospitality of Prof. Frank E. Harris during visits to Quantum Theory Project, University of Florida. This work has been supported by the National Science Centre, Poland under Grant No. 2013/11/B/ST4/02191.

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