CASPT2 and FS-MR-CC

Article pubs.acs.org/JCTC

Potential Energy Curves in the CASSCF/CASPT2 and FS-MR-CC Methods: The Role of Relativistic Effects Maria Barysz Department of Quantum Chemistry, Faculty of Chemistry, Nicolaus Copernicus University, Gagarina 7, 87-100 Toruń, Poland ABSTRACT: Ab initio CASSCF/CASPT2 calculations for the electronic ground and for a wide range of excited states of Li2 and Na2 dimers are presented. The computed spectroscopic parameters agree very well with the experimental data. This indicates that the old CASSCF/CASPT2 method can be as successfully applied to study excited states of molecules as recently developed the multireference Fock-space coulped-cluster method. The role of relativistic effects in the correct description of the potential energy curves has been investigated using as an example the SiAu molecule. The accuracy of the new infinite-order two-component relativistic method has been studied and its advantage over the Douglas−Kroll−Hess method demonstrated.

1. INTRODUCTION The electronic structures of the ground states of many molecules are described by a wave function dominated by a single configuration. The situation is different for excited states, where the energy separation between different electronic configurations is small. This results in strong mixing of the states. Consequently, the excited molecular states have to be treated using multiconfigurational (MC) theories.1 The recent improvement of methods and techniques of ab initio quantum chemistry has enlarged the possibilities of obtaining accurate theoretical information about spectroscopic properties of molecular systems in their excited states. In particular, the development of the coupled cluster (CC) MC approaches has increased the ability to obtain results with accuracy comparable to that of the experimental data.2−6 To find the spectroscopic parameters of electronic states, one needs to create the appropriate potential energy curve (PEC). Although the coupled cluster methods are among the most accurate in quantum chemistry, the necessity to use a singledeterminant Hartree−Fock function as the starting point in any CC calculation makes them difficult to use for the description of bond dissociation. The restricted Hartree−Fock (RHF) theory is not able to properly described the open-shell structures, whereas the unrestricted HF (UHF) suffers from symmetry breaking. The breakthrough was created by the formulation and implementation of the multireference coupled cluster theory within the Fock space formalism (FS-MR-CC), especially for the so-called (0,2) and (2,0) sectors, which do not suffer from the HF disadvantages. The N-electron RHF-based FS-MR-CC method, applied to a system that dissociates to two closed shell subsystems, enables calculation of the energy of N − 2 ((0,2) sector) or N + 2 ((2,0) sector) electronic systems, which dissociate to two open shell subsystems. Because of rigorous size-consistency, this allows for obtaining correctly behaving dissociation curves. The performance of the FS-MR-CC method has been illustrated by numerous examples of correctly calculated spectroscopic parameters and potential energy curves of excited states of diatomic structures.5,7−10 © 2016 American Chemical Society

It seems, however, that we have quite forgotten discussions about old multireference complete active space self-consistent field (CASSCF) and complete active space second-order perturbation theory (CASPT2) methods formulated by Roos and his co-workers.1,11−14 The generality of the CASSCF/ CASPT2 approach makes them especially suitable for studies of excites states of atoms and molecules. The CASSCF wave function is defined by selecting a set of active orbitals and is constructed as a linear expansion in the set of configuration functions that can be generated by occupying the active orbitals in all ways consistent with an overall spin and space symmetry. No assumptions are made regarding the character of the excited states (singly or multiply excited) or the shapes of the molecular orbitals. The latter are optimized with the only limitations set by the chosen basis set (in comparison, the FS-MR-CC method does not optimize the molecular orbitals and uses the HF orbitals as the same orbitals for all excited states). The CASSCF method can handle the near-degeneracy problem in a balanced and efficient manner, but it does not include the effect of the dynamic correlation. The missing correlation is here included using the second order perturbation theory (PT2) with the multireference CASSCF wave function. The perturbation CASPT2 method is also size-consistent and can therefore be applied to systems with many electrons without loss of accuracy. Provided that extended basis sets are used and the active space of the CASSCF reference function is properly chosen, the CASPT2 method gives highly accurate energies for a broad range of chemical problems.15−21 Because of the general nature of the reference function, the CASSCF/CASPT2 method should work equally well for ground and excited states and for electronic states that are not wellcharacterized by a single configuration (cited from ref 13). However, I should add some additional comments. The CASPT2 method is based on second order perturbation theory. For the method to be successful, the perturbation should be small. A correct selection of the active space in the preceding Received: January 5, 2016 Published: February 25, 2016 1614

DOI: 10.1021/acs.jctc.6b00008 J. Chem. Theory Comput. 2016, 12, 1614−1626

Article

Journal of Chemical Theory and Computation

All calculations are carried out using the MOLCAS7.3 package of quantum chemistry program.22,30 The spectroscopic parameters result from a rovibronic analysis of the calculated CASSCF/ CASPT2 potential energy curves and have been obtained by the Numerov−Cooley method and the VIBROT program of the MOLCAS7.3 package.

CASSCF calculation is therefore of utmost importance. All neardegeneracy effects leading to configurations with large weights must be included at this stage of the calculations; otherwise, the first order wave function will contain large coefficients. When this occurs, the CASPT2 program issues a warning. If the energy contribution from such a configuration is large, a new selection of the active space should be made; this is how it is done for some Π states in my work. In calculations on excited states, intruder states occur in the first order wave function. Warnings are then used by the program that an energy denominator is small or negative. Such intruder states often arise from Rydberg orbitals, which have not been included in the active space. Even if this sometimes leads to large CI coefficients, the contribution to the second order energy is usually very small because the interaction with the intruder Rydberg state is small. It might be safe to neglect the warning. A safer procedure is to include the Rydberg orbital in the active space, whereas it can sometimes be deleted from the MO space (cited from the Molcas7.3 manual, ref 22). According to my knowledge, the CASSCF/CASPT2 methods have never been used to systematically calculate a large series of spectroscopic constants of a single molecular system, as was recently done by Musial et al.9 One of the goals of the current research is to demonstrate the performance of the CASSCF/ CASPT2 method for some difficult cases and to compare the obtained results with the FS-MR-CC values.9 In the current work, I generate the potential energy curves for the ground and multiple excited states of the alkali metal dimers Li2 and Na2. In the FS-MR-CC methodology, the idea relies on being able to perform the RHF-based CC calculations for the double positive ions of Li2+2 or Na2+2. The Li2+2 and Na2+2 systems dissociate into Li+ and Na+ positive ions, which are closed shell units. Through FS-MR-CC (2,0) calculations, the wave functions for the neutral systems Li2 and Na2 are recovered. Compared to the FS-MR-CC method, the CASSCF/CASPT2 method is not affected by the problem of the wrong dissociation in the RHF methodology. Moreover, the RHF molecular orbitals can be used for an arbitrary ion and geometry of the studied system. In the following CASSCF/CASPT2 calculations, the CASSCF orbitals and integrals from the earlier geometry can be used. Nature is relativistic. One can not obtain a correct potential energy curve without taking into account the relativistic effects. The second goal of the present work is to show the importance of the relativistic effects in the proper description of the potential curves and their spectroscopic parameters and to test the new relativistic infinite-order two-component (IOTC) method. Although both methods, nonrelativistic and relativistic IOTC, have been used for the calculations of the spectroscopic parameters of diatomic Li2 and Na2 molecules, these calculations do not show the role of the relativistic effects. The Li2 and Na2 molecules contain light atoms, and the role of relativistic effects here is negligible. The IOTC calculations for these systems show, however, how the IOTC method smoothly converges to the nonrelativistic case. For showing how the spectroscopic parameters change when relativistic effects are large, the SiAu molecule containing heavy element Au has been studied. Calculations by the IOTC FS-MR-CC method are currently not possible, and I report IOTC CASSSCF/CASPT2 results only. All calculations have been carried out by the relativistic IOTC method formulated by the author.23−29 The performance of the IOTC method has been tested, and a comparison with the relativistic Douglas−Kroll−Hess (DKHn) approximation is presented.

2. COMPUTATIONAL METHODOLOGY: LI2 DIATOMIC MOLECULE All calculations have been carried out with the complete active space CASSCF method followed by the second-order singlestate multireference perturbation CASPT2 scheme. Two basic active spaces have been used in the calculations of the Li2 molecule. The first active space consists of the 1s, 2s, and 2p atomic orbitals of Li atom. The second active space consists of 1s, 2s, 3s, 2p, and 3p orbitals. All calculations have been carried out in D2h symmetry. Thus, the orbital (frozen/inactive/active) subspaces are defined by the number of orbitals in irreducible representations of that group (ag, b3u, b2u, b1g, b1u, b2g, b3g, au). The partition of the orbitals space used in CASSCF and CASPT2 calculations is then (00000000/00000000/31103110; nel) and (00000000/00000000/52205220; nel) for the first and second active space, respectively, where nel is the number of electrons in the active space (6 for the Li2 dimer calculations). For some Δ states, which might interfere with the Π state, to be checked, some pilot calculations in a lower C2 symmetry have been performed. No degeneracy of Π and Δ states have been observed. The CAS wave function is a full CI wave function in the space of active orbitals. The CASSCF wave function has been obtained by optimizing both CI coefficients and the active and inactive orbitals. The choice of the basis set is one of the most important steps in all calculations. It is particularly important in calculations of excited states. However, the aim of the current study is to compare the published FS-MR-CC results with the calculated CASSSCF/CASPT2 ones. This means that I have to use the basis sets identical to the ones used earlier in the FS-MR-CC calculations.9 The extended basis sets used in the recent FS-MRCC calculations10 cause some linear dependencies in the CASSCF/CASPT2 calculations. Any further analysis of the impact of the basis set effects on the received spectroscopic parameters has not been carried out in this work. Two basis sets have been used, the nonrelativistic contracted Sadlej’s POL basis set [10s.6p.4d/5s.3p.2d] for the CASSCF/ CASPT2 calculations 31 and the contracted ANO-RCC [14s9p4d3f1g/8s7p4d2f1g] basis for the relativistic IOTC CASSCF/CASPT2 calculations.32 In the corresponding FSMR-CC calculations, with the POL basis set, 5 lowest virtual orbitals have been selected as the active ones. This is comparable to both (21102110) and (31103110) NR CASSCF/CASPT2 active spaces. In the FS-MR-CC calculations with the contracted ANO-RCC basis, the 25 lowest virtual orbitals have been used as active. This space is a little bigger than the one used in (52205220) CASSSCF/CASPT2 calculations. All FS-MR-CC calculations have been performed for the nonrelativistic Hamiltonian using two basis sets, POL or ANORCC. There is some problem, however, with the use of the contracted ANO-RCC basis. The original contracted ANO-RCC basis sets were optimized for the relativistic Hamiltonian, and they should not be used within the nonrelativistic scheme, as it was done in the FS-MR-CC calculations. It seems, however, that for light elements, such as Li or Na atoms, for whom the 1615


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+ Li(2p) (six states), and Li(2s) + Li(3s) (three states). All curves are smooth with no qualitative difference compared to the FS-MR-CC curves published in ref 9. Even in very difficult cases, such as for the 31Σ+g and 21Σ+u states, when a double minimum occurs on the potential energy curve, the CASSCF/CASPT2 method gives a very good representation of the curves. The computed spectroscopic constants are listed in Tables 3−8 for the POL (Tables 3−5) and for the ANO-RCC (Tables 6−8) basis sets. I see that the calculated values of Re, De, and ωe agree very well with the experimental data for all states and in both basis sets. The agreement improves when the larger (52205220) active space is applied (for 11Πu state (73317331) active space). For (522052220) active space, the equilibrium geometry is off by 0.01−0.05 Å in the POL basis and by 0.0−0.02 Å in ANO RCC. The exceptions are 11Πg and 21Σ+u states calculated in the ANO RCC basis for which the discrepancy is 0.08 and 0.05 Å, respectively. Similarly, the dissociation energy De is off by 0.01−0.05 eV in the POL basis (only for the 11Πu state is the error larger at 0.07 eV) and by 0.0−0.04 eV in the ANO RCC basis. The exceptions are states that dissociate into the Li(2s) + Li(3s) limit calculated in the ANO RCC basis. Here, the dissociation energy is off by approximately 0.10−0.18 eV. For these highly excited states, the POL basis gives much better results. Similarly, the errors in the harmonic frequency ωe do not exceed 10 cm−1 in the POL basis or 5 cm−1 in the ANO RCC basis for most of the cases. As was mentioned, the potential energy curves of the calculated states presented in Figure 1 look quite good in comparison to the curves obtained in the FS-MR-CC method. The quality of the curves can also be assessed by the calculation of the ωexe parameter. The calculation of this parameter is much more demanding than the calculations of others spectroscopic constants. Nevertheless, the presented values of ωexe are quite good. In some cases, my data are further from the exact values, but this is also the case in the FS-MR-CC approach. Most difficult are the states that dissociate to the 2s + 3s limit and have two minima on the curve. However, one should mention here that the Molcas Vibrot program used to compute spectroscopic constants is not able to treat such curves and that parameters for 31Σ+g and 21Σ+u states have only approximate values. Very problematic also is the value of the ωexe constant for the 21Σ+g state. The calculated CASSSCF/CASPT2 value is negative, whereas the FS-MR-CC ωexe parameter is positive. Similarly, two different (negative and positive) experimental values have been reported. However, I do not see any “spikes” or irregularities on my 21Σ+g potential energy curve that could be blamed for this result. Although at an intermediate internuclear distance most of the investigated states have a multiconfigurational character, in the vicinity of the equilibrium distance, one finds for all of them at most two dominant configurations. These dominant configurations at the equilibrium bond distance Re of all considered states are listed in Table 9. In Tables 10 and 11, I present the adiabatic excitation energies calculated in POL and ANO-RCC bases. The calculated energies agree very well with both FS-MR-CC and experimental values. Finally, Table 12 illustrates the size-consistency of the CASSCF/CASPT2 method. If a theory for the evaluation of energy is size consistent, then the energy of the supersystem A + B, separated by a large distance, is equal to the sum of the energy of A plus the energy of B, in my case, E(Li + Li) = E(Li) + E(Li). The property of size consistency is of particular importance to obtain a correctly behaving dissociation curve. The energy values

relativistic effects are small, it is irrelevant if I use the relativistic, contracted ANO-RCC basis in either nonrelativistic or relativistic calculations. For the accuracy of this assumption to be shown, nonrelativistic and IOTC relativistic calculations have been performed in an uncontracted ANO-RCC basis for the ground X1Σ+g state and 13Σ+g excited state of the Li2 dimer and for the ground X1Σ+g state and 13Σ+u and 13Σ+g states of the Na2 dimer. The analysis of the data in Tables 1 and 2 shows that the Table 1. Spectroscopic Constants for the Li2 Molecule in the Relativistic IOTC and Nonrelativistic (NR) Methods and the Uncontracted ANO-RCC Basis Set method

Re (Å)

De (eV)

ωe (cm−1)

ωexe (cm−1)

active space

2s + 2s Dissociation Limit X1Σ+g IOTC CASSCF/ CASPT2 NR CASSCF/ CASPT2

2.68

1.04

346

2.52

(21102110)

2.68

1.04

346

2.52

(21102110)

2s + 2p Dissociation Limit 13Σ+g IOTC CASSCF/ CASPT2 NR CASSCF/ CASPT2

3.09

0.86

248

2.36

(52205220)

3.09

0.86

248

2.34

(52205220)

Table 2. Spectroscopic Constants for the Na2 Molecule in the Relativistic IOTC and Nonrelativistic (NR) Methods and the Uncontracted ANO-RCC Basis Set method

Re (Å)

De (eV)

ωe (cm−1)

ωexe (cm−1)

active space

3s + 3s Dissociation Limit X1Σ+g IOTC CASSCF/ CASPT2 NR CASSCF/ CASPT2 13Σ+u IOTC CASSCF/ CASPT2 NR CASSCF/ CASPT2

3.09

0.74

158

0.60

(21102110)

3.09

0.74

158

0.60

(21102110)

5.17

0.02

25

0.68

(21102110)

5.16

0.02

25

0.68

(21102110)

3s + 3p Dissociation Limit 13Σ+g IOTC CASSCF/ CASPT2 NR CASSCF/ CASPT2

3.80

0.58

100

0.58

(21102110)

3.80

0.58

100

0.58

(21102110)

nonrelativistic NR CASSCF/CASP2 and relativistic IOTC CASSCF/CASPT2 calculations give the same values of all spectroscopic parameters. This confirms the validity of my assumption. In this work, to be consistent with the theory, the contracted ANO-RCC basis set is always used in the relativistic IOTC CASSCF/CASPT2 calculations.

3. LITHIUM DIMER: RESULTS AND DISCUSSION In Figure 1, a set of potential energy curves corresponding to the ground and to ten excited states of the Li2 molecule as a function of the intermolecular distance obtained with the POL basis set and (52205220) active space is presented. The curves shown in Figure 1 represent states dissociating to Li atoms in three dissociation limits: Li(2s) + Li(2s) (two molecular states), Li(2s) 1616


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Figure 1. Potential energy curves for the ground and excited states of the Li2 molecule with the CASSCF/CASPT2 method for Li(2s) + Li(2s), Li(2s) + Li(2p), and Li(2s) + Li(3s) dissociation limits in the POL basis set and the (52205220) active space.

4. THE GROUND AND EXCITED STATES OF THE SODIUM DIMER: RESULTS AND DISCUSSION In Figure 2, potential energy curves for the ground and excited states of the sodium dimer in the IOTC CASSCF/CASPT2 method for the 3s + 3s and 3s + 3p dissociation limits in the ANO-RCC basis set are presented. In Figure 3, potential energy curves for the excited states of the sodium dimer in the CASSCF/CASPT2 method for the 3s + 4s dissociation limits in the POL basis set are presented. Just as it was for the Li2 dimer, for the Na2 molecule I also observe smooth curves similar to those obtained in the FS-MRCC method.9 Most of the calculations have been carried out with the relativistic IOTC CASSCF/CASPT2 method and the ANORCC basis [17s12p5d4f/9s8p5d4f].32 For the states that dissociate into the 3s + 4s limit, both ANO-RCC and POL basis sets have been used. The reason is that the spectroscopic constants computed in the ANO-RCC basis do not always give satisfactory results in both the FS-MR-CC and IOTC CASSCF/ CASPT2 methods. Calculations have been carried out in D2h symmetry. The partition of the orbital space used in D2h symmetry is (00000000/31103110/21102110; 2el) in the CASSCF method. This means that the 3s, 3p active (2 el/8 orb) has been used in the CASPT2 method. For the highly excited states, which dissociate into the 3s + 4s limit, the partition is (00000000/31103110/42204220; 2el) and means that 3s, 3p and 4s, 4p orbitals have been chosen as active. In the CASPT2 calculations, the core 1s orbitals are kept frozen in all calculations. The active space and the number of active electrons in the CASSCF method are not large. The major part of the correlation energy has been calculated at the CASPT2 level where all excitations from the active and inactive orbitals are allowed. The CASSCF/CASPT2 active space used in my calculations has been suggested by the Molcas authors.22 All near-degeneracy effects leading to configurations with large weights have been included at the CASSCF level.

Table 3. Spectroscopic Constants for the Li2 Molecule in the POL Basis for the 2s + 2s Dissociation Limit method

Re (Å)

De (eV)

ωe (cm−1)

ωexe (cm−1)

active space

2s + 2s Dissociation Limit X1Σ+g CASSCF/ CASPT2 CASSCF/ CASPT2 FS-MR-CCc expta 13Σ+u CASSCF/ CASPT2 CASSCF/ CASPT2 FS-MR-CCc exptb

2.70

1.08

339

2.42

(31103110)

2.70

1.02

343

2.36

(52205220)

2.64 2.67

1.13 1.06

370 351

2.98 2.58

4.19

0.13

70

1.22

(31103110)

4.14

0.04

69

4.44

(52205220)

3.82 4.17

0.08 0.04

84 65

2.89 3.27

a Experimental data from refs 40 and 41. bExperimental data from refs 42 and 43. cFS-MR-CC from ref 9.

show that the method is size consistent and, more importantly, that the dissociation of the calculated curves is correct. Recently, Musial et al.10 published results of more extensive FS-MR-CC calculations with a modified uncontracted ANORCC basis and with much greater active space than the one used in the present work. The model space in this report contains 86 lowest virtual orbitals, and the resulting size of the active space (2,0) is equal to 7396 determinants. The published results for the lithium dimer excited states converge almost exactly to the experimental data, including the shape of the potential energy curves. It should be emphasized, however, that the CASSCF/ CASPT2 method gives very good results for a relatively small active space and small basis, and this was the goal I wanted to achieve. 1617


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Journal of Chemical Theory and Computation Table 4. Spectroscopic Constants for the Li2 Molecule in the POL Basis for the 2s + 2p Dissociation Limit state 13Πu

11Σ+u

13Σ+g

11Πu

21Σ+g

11Πg

method CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg expta CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg exptb CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg exptc CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg exptd CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg expte CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCg exptf

Re (Å) 2.62 2.62 2.56 2.59 3.12 3.14 3.06 3.11 3.07 3.12 3.02 3.07 3.02 3.01 2.91 2.94 3.64 3.64 3.48 3.65 4.24 4.10 3.88 4.06

De (eV)

ωe (cm−1)

2s + 2p Dissociation Limit 1.45 1.48 1.60 1.51 2.16 1.15 1.22 1.17 0.90 0.87 0.97 0.88 0.26 0.30 0.44 0.37 0.40 0.38 0.46 0.41 0.16 0.17 0.19 0.18

339 340 367 346 263 255 269 257 272 251 254 251 260 261 291 271 136 132 139 129 121 93 98 93

ωexe (cm−1)

active space

1.95 1.91 2.33 1.89 1.56 1.72 1.71 1.58 1.95 2.46 2.54 2.35 3.01 2.96 4.13 2.95 0.33 0.52 −0.48 1.70 (−1.48h) 2.97 2.82 1.53 1.87

(31103110) (52205220)

(31103110) (52205220)

(31103110) (52205220)

(31103110) (52205220)

(31103110) (52205220)

(31103110) (52205220)

a Experimental data from ref 44. bExperimental data from ref 45. cExperimental data from ref 43. dExperimental data from ref 40. eExperimental data from ref 46. fExperimental data from ref 47. gFS-MR-CC from ref 9. hExperimental data from ref 48.

Table 5. Spectroscopic Constants for the Li2 Molecule in the POL Basis for the 2s + 3s Dissociation Limit state 23Σ+g

31Σ+g

31Σ+g 21Σ+u

21Σ+u

method CASSCF/CASPT2 FS-MR-CCd expta CASSCF/CASPT2 inner FS-MR-CCd exptb CASSCF/CASPT2 exptf CASSCF/CASPT2 FS-MR-CCd exptc CASSCF/CASPT2 FS-MR-CCd exptc

Re (Å) 3.12 3.05 3.09 3.14 3.06 3.09 5.92 5.74 3.13 3.06 3.10 6.14 6.08 6.04

De (eV) 2s + 3s Dissociation Limit 1.00 1.09 0.98 1.07 1.03 0.72 0.72 0.70 0.77 0.70 0.66 0.70 0.66

ωe (cm−1)

ωexe (cm−1)

270 278 269 232 241 246

1.46 1.42 1.50 5.23 4.13 2.83

active space (52205220)

(52205220) inner

(52205220) outer 258 273 259 117 117

1.99 2.67 1.71 0.74 0.60

(52205220) inner

(52205220) outer

a Experimental data from ref 49. bExperimental data from ref 50. cExperimental data from refs 51 and 52. dFS-MR-CC from ref 9. fExperimental data from ref 53.

In the cited FS-MR-CC data, 25 virtual orbitals were used as active.9 To check that there are no close lying Δ and Π states, I have performed pilot calculations in C2 symmetry as well. I did not observe degeneracy of these states. The dominant configurations at the equilibrium bond distance Re of all considered states are listed in Table 13. The computed spectroscopic parameters for the states that dissociate to 3s + 3s and 3s + 3p atomic limits are listed in Tables 14 and 15. The IOTC CASSCF/CASPT2 equilibrium bond distances, the dissociation energies, and the harmonic

frequencies correlate very well with the experimental data. The equilibrium distance Re is off by 0.0−0.04 Å; for the 13Σ+u state alone, the error is 0.07 Å. For comparison, the FS-MR-CC distance for this state is 0.19 Å off of the experimental value. The calculated dissociation energies De are also very good and differ from the experimental values by 0.0−0.05 eV. The error of the calculated harmonic frequencies ωe is not larger than 5 cm−1. The calculated ωexe constant is good for most of the states, except 11Πu and 11Πg, for which the errors are 0.33 and 0.39 cm−1, respectively. It has been shown already for the Li2 molecule that the Π states are more difficult to compute and require much 1618


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Journal of Chemical Theory and Computation Table 6. Spectroscopic Constants for the Li2 Molecule in the ANO-RCC Basis for the 2s + 2s Dissociation Limit Re (Å)

method

ωe (cm−1)

De (eV)

ωexe (cm−1)

active space

(21102110) CONTR. (21102110) UNCONTR. (52205220) CONTR. (74417441) UNCONTR.

2s + 2s Dissociation Limit X1Σ+g CASSCF/CASPT2 CASSCF/CASPT2 CASSCF/CASPT2 CASSCF/CASPT2 FS-MR-CCc expta 13Σ+u CASSCF/CASPT2 FS-MR-CCc exptb a

2.68 2.68 2.68 2.68 2.68 2.67

1.04 1.04 1.05 1.04 1.05 1.06

346 346 351 350 351 351

2.52 2.52 2.52 2.50 2.71 2.58

4.17 4.16 4.17

0.04 0.04 0.04

67 65 65

2.89 3.27

(52205220)

Experimental data from refs 40 and 46. bExperimental data from refs 42 and 43. cFS-MR-CC from ref 9.

Table 7. Spectroscopic Constants for the Li2 Molecule in the ANO-RCC Basis for the 2s + 2p Dissociation Limit state 13Πu

11Σ+u

13Σ+g

11Πu

21Σ+g

11Πg

Re (Å)

method IOTC CASSCF/CASPT2 FS-MR-CCg expta IOTC CASSCF/CASPT2 FS-MR-CCg exptb IOTC CASSCF/CASPT2 FS-MR-CCg exptc IOTC CASSCF/CASPT2 IOTC CASSCF/CASPT2 IOTC CASSCF/CASPT2 FS-MR-CCg exptd IOTC CASSCF/CASPT2 FS-MR-CCg expte IOTC CASSCF/CASPT2 FS-MR-CCg exptf

2.60 2.59 2.59 3.11 3.11 3.11 3.09 3.07 3.07 2.96 2.95 2.94 2.94 2.94 3.66 3.66 3.65 4.14 4.06 4.06

De (eV) 2s + 2p Dissociation Limit 1.47 1.49 1.51 1.16 1.15 1.17 0.86 0.87 0.88 0.30 0.33 0.36 0.37 0.37 0.38 0.42 0.41 0.16 0.17 0.18

ωe (cm−1)

ωexe (cm−1)

active space

345 346 346 254 255 257 248 254 251 265 266 269 271 271 126 130 129 88 93 93

1.91 1.95 1.89 1.28 1.54 1.58 2.48 2.31 2.35 3.00 2.99 2.91 3.51 2.95 0.79 −1.10 1.70−1.48h 2.25 1.66 1.87

(52205220)

(52205220)

(52205220)

(52205220) (53305330) (73317331)

(52205220)

(52205220)

a

Experimental data from ref 44. bExperimental data from ref 45. cExperimental data from ref 43. dExperimental data from ref 40. eExperimental data from ref 46. fExperimental data from ref 47. gFS-MR-CC from ref 9. hExperimental data from ref 48.

Table 8. Spectroscopic Constants for the Li2 Molecule in the ANO-RCC Basis for the 2s + 3s Dissociation Limit state 23Σ+g

31Σ+g

21Σ+u

21Σ+u

a

method IOTC CASSCF/CASPT2 FS-MR-CCd expta IOTC CASSCF/CASPT2 FS-MR-CCd expta CASSCF/CASPT2 FS-MR-CCd exptb IOTC CASSCF/CASPT2 FS-MR-CCd exptc

Re (Å) 3.08 3.08 3.09 3.19 3.08 3.09 3.10 3.07 3.10 6.09 6.09 6.04

ωe (cm−1)

De (eV)

2s + 3s Dissociation Limit 1.05 276 1.03 270 269 1.21 235 1.01 235 1.03 246 0.85 257 0.63 258 0.70 259 0.77 117 0.68 118 0.66

ωexe (cm−1) 2.05 1.42 1.50


(52205220) inner 5.14 3.27 (52205220) inner UNCONTR. 1.72 (52205220) outer UNCONTR. 0.51

Experimental data from ref 54. bExperimental data from ref 50. cExperimental data from refs 51 and 52. dFS-MR-CC from ref 9.

1619


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Journal of Chemical Theory and Computation Table 9. Dominant Configurations in the Equilibrium Distance for the Li2 Molecules in the CASSCF/CASPT2 Calculations and POL Basis in the (52205220) Active Space symmetry

weight

X1Σ+g 13Σ+u

0.90 0.98

13Πu 11Σ+u 13Σ+g 11Πu 21Σ+g

0.95 0.92 0.96 0.93 0.54, 0.26

11Πg

0.59, 0.37

23Σ+g 21Σ+u 21Σ+u 31Σ+g

0.87 0.91 0.72 0.75

Table 11. Adiabatic Excitation Energies for the Li2 Molecule in the POL and ANO-RCC Basis Sets for the 2s + 3s Dissociation Limit

configuration 2s + 2s Dissociation Limit (1σg)2(1σu)2(2σg)2 (1σg)2(1σu)2(2σg)1(2σu)1 2s + 2p Dissociation Limit (1σg)2(1σu)2(2σg)1(1πxu)1 (1σg)2(1σu)2(2σg)1(2σu)1 (1σg)2(1σu)2(2σg)1(3σg)1 (1σg)2(1σu)2(2σg)1(1πxu)1 (1σg)2(1σu)2(2σg)1(3σg)1 + (1σg)2(1σ*u )2(2σ*u )2 (1σg)2(1σu)2(2σu)1(1πxu)1 + (1σg)2(1σu)2(2σg)1(1πxg)1 2s + 3s Dissociation Limit (1σg)2(1σu)2(2σg)1(3σg)1 (1σg)2(1σu)2(2σg)1(3σu)1 (1σg)2(1σu)2(2σg)1(3σu)1 (1σg)2(1σu)2(2σg)1(3σg)1

state 13Σ+u

1 Πu 3

11Σ+u

13Σ+g

11Πu

21Σ+g

11Πg

method

POL (eV)

2s + 2s Dissociation Limit CASSCF/CASPT2 0.98 0.97 FS-MR-CCb 1.06 1.01 expta 1.01 1.01 2s + 2p Dissociation Limit CASSCF/CASPT2 1.36 1.39 FS-MR-CCb 1.37 1.39 expta CASSCF/CASPT2 1.69 1.70 FS-MR-CCb 1.75 1.74 expta 1.74 1.74 CASSCF/CASPT2 1.97 2.01 FS-MR-CCb 2.03 2.02 expta 2.02 2.02 CASSCF/CASPT2 2.54 2.56 FS-MR-CCb 2.57 2.54 expta 2.53 2.53 CASSCF/CASPT2 2.46 2.48 FS-MR-CCb 2.55 2.49 expta 2.49 2.49 CASSCF/CASPT2 2.66 2.68 FS-MR-CCb 2.78 2.72 expta 2.73 2.73

method

23Σ+g

CASSCF/ CASPT2 FS-MR-CCb expta CASSCF/ CASPT2 FS-MR-CCb expta CASSCF/ CASPT2 FS-MR-CCb expta

21Σ+u

21Σ+u

POL (eV)

ANO-RCC (eV)

2s + 3s Dissociation Limit 3.47 3.50 3.44


3.41

3.66

(52205220) inner

3.75 3.73 3.69

3.81 3.73

3.82 3.77

3.75 3.75

(52205220) outer

a

Experimental data from refs 40, 45, 50, and 55. bFS-MR-CC from ref 9.

inner outer inner

Table 12. Energy in the Dissociation Limit for the Li2 Molecules in the CASSCF/CASPT2 Calculations and POL Basis Sets in the (52205220) Active Space

Table 10. Adiabatic Excitation Energies for the Li2 Molecule in the POL and ANO-RCC Basis Sets ANO-RCC (eV)

state

molecular symmetry active space X1Σ+g 13Σ+u 2s + 2s Li atom

(52205220)

13Πu 11Σ+u 13Σ+g 11Πu 21Σ+g 11Πg 2s + 2p Li atom

(52205220)

(52205220)

(52205220)

23Σ+g 31Σ+g 21Σ+u 31Σ+g 2s + 3s Li atom

(52205220)

a

(52205220)

energy (au) 2s + 2s Dissociation Limit −14.8819228 −14.8819228 −14.8819228 2s + 2p Dissociation Limit −14.815068 −14.815058 −14.815079 −14.815049 −14.815072 −14.815064 −14.815057 2s + 3s Dissociation Limit −14.759497 −14.759497 −14.759493

ΔEa 0.0000000 0.0000000

0.000011 0.000001 0.000022 0.000008 0.000015 0.000007

0.000009 0.000009 0.000013

−14.759506

ΔE = E(Li2) − E(Li) − E(Li) (atoms in the dissociation limit).

(52205220)

results than the ANO-RCC basis in both the FS-MR-CC and CASSSCF/CASPT2 methods. It is interesting to note the strong dependence of the results on the basis set used. The same applies to both the FS-MR-CC and CASSCF/CASPT2 methods. I have not received good ωexe for the highest excited state 21Σ+u in the ANO-RCC basis, and they are not presented in the table. Finally, it is noteworthy to add that, although the Vibrot Molcas program did not catch the second minimum on the 31Σ+g curve, it shows its presence on the potential curve presented in the Figure 3. Tables 17 and 18 provide the calculated adiabatic excitation energies for all examined states. Again, I observe a very good agreement between CASSCF/CASPT2 and the experimental energies. Finally, Table 19 illustrates the size-consistency of the CASSCF/CASPT2 method and the proper dissociation of the calculated curves.

a

Experimental data from ref 40, 45, 50, and 55. bFS-MR-CC from ref 9.

larger active space in the CASSCF calculations than for the other excited states. The FS-MR-CC ωexe parameters of the Π states are better then those obtained in the CASSCF/CASPT2 method. On the contrary, the CASSCF/CASPT2 method gives much better results than the FS-MR-CC method for the 13Σ+u state. The corresponding potential energy curves are good in both methods. In Table 16, results of calculations for the states that dissociate to the 3s + 4s limit in the ANO-RCC and POL basis sets are presented. It can be observed that the POL basis gives better 1620


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Figure 2. Potential energy curves for the ground and excited states of the Na2 molecule in the IOTC CASSCF/CASPT2 method for Na(3s) + Na(3s) and Na(3s) + Na(3p) dissociation limits in the ANO-RCC basis set and (21102110) active space.

Figure 3. Potential energy curves for the excited states of the Na2 molecule in the CASSCF/CASPT2 method for the Na(3s) + Na(4s) dissociation limits in the POL basis set and (42204220) active space.

of this paper, I present some results for the “relativistic” SiAu molecule and compare them to the nonrelativistic data. The final goal of the paper is to show that my new relativistic IOTC method works well and can be successfully used in a study of the excited states of molecules. For relatively light elements, the electronic structure of their compounds may be studied in terms of solutions of the nonrelativistic Schrödinger equation. With the increasing value of the nuclear charge, the relativistic effects become increasingly important. As a consequence, the Schrödinger Hamiltonian

V. THE ROLE OF RELATIVISTIC EFFECTS IN DETERMINING THE POTENTIAL ENERGY CURVES The main goal of the research presented in this paper is to search for a good method to describe the potential energy curves of molecules and their spectroscopic parameters. The results presented in the first part of this work have been obtained for the light elements only, with simple valence electronic structure, for which the relativistic effects are small. However, in general, only the energy curves containing relativistic effects can be compared with the experimental data. This is why, in the last part 1621


Article

Journal of Chemical Theory and Computation Table 13. Dominant Configurations of the Equilibrium Distance for the Na2 Molecules in the CASSCF/CASPT2 Calculations in the (31103110) Inactive and (21102110) Active Spaces symmetry

weight

X1Σ+g 13Σ+u

0.91 0.99

1 Πu 11Σ+u 13Σ+g 11Πu 21Σ+g 11Πg

0.98 0.94 0.98 0.94 0.36, 0.40 0.61, 0.38

23Σ+g 21Σ+u 21Σ+u

0.96 0.95 0.46, 0.14, 0.14 0.52, 0.27

3

31Σ+g

Table 15. Spectroscopic Constants for the Na2 Molecule in the ANO-RCC Basis for the 3s + 3p Dissociation Limit in the CASSCF (31103110) Inactive Space state

configuration 3s + 3s Dissociation Limit [Ne2](4σg)2 [Ne2](4σg)1 (4σu)1 3s + 3p Dissociation Limit [Ne2](4σg)1(2πxu)1 [Ne2](4σg)1(4σu)1 [Ne2](4σg)1(5σg)1 [Ne2](4σg)1 (2πxu)1 [Ne2](4σg)1(5σg)1 + [Ne2](4σu)2 [Ne2](4σg)1 (2πxg)1 + [Ne2](4σu)1(2πxu * )1 3s + 4p Dissociation Limit [Ne2](4σg)1(6σg)1 [Ne2](4σg)1(5σu)1 [Ne2](4σg)1(4σu)1 + [Ne2](4σg)1 (5σu)1 + [Ne2](5σg)1 (4σu)1 [Ne2](4σg)1(5σg)1 + [Ne2](4σg)1 (6σg)1

13Πu

11Σ+u

13Σ+g

11Πu

inner outer

Table 14. Spectroscopic Constants for the Na2 Molecule in the ANO-RCC basis for the 3s + 3s Dissociation Limit in the CASSCF (31103110) Inactive Space method

Re (Å)

De (eV)

ωe (cm−1)

ωexe (cm−1)

21Σ+g

11Πg

active space

3s + 3s Dissociation Limit X1Σ+g IOTC CASSCF/ CASPT2 FS-MR-CCc expta 13Σ+u CASSCF/CASPT2 FS-MR-CCc exptb

3.08

0.75

158

0.61

3.06 3.08

0.78 0.75

156 159

0.65 0.72

5.16 4.90 5.09

0.02 0.03 0.02

25 31 24

0.67 0.85 0.64

method

Re (Å)

De (eV)

ωe (cm−1)

3s + 3p Dissociation Limit IOTC CASSCF/ 3.11 1.15 156 CASPT2 FS-MR-CCf 3.09 1.20 152 3.12 1.17 154 expta IOTC CASSCF/ 3.65 1.03 117 CASPT2 3.63 1.05 118 FS-MR-CCf 3.64 1.03 112 exptb IOTC CASSCF/ 3.82 0.57 99 CASPT2 3.78 0.60 101 FS-MR-CCf 3.78 0.59 101 exptc IOTC CASSCF/ 3.43 0.27 265 CASPT2 IOTC CASSCF/ 3.43 0.30 266 CASPT2 3.43 0.35 271 FS-MR-CCf 3.43 0.33 271 exptd IOTC CASSCF/ 4.46 0.44 76 CASPT2 FS-MR-CCf 4.41 0.47 75 4.45 0.45 75 expte IOTC CASSCF/ 4.59 0.14 45 CASPT2 FS-MR-CCf 4.56 0.16 47 4.56 0.15 43 expte

ωexe (cm−1)

active space

0.46

(21102110)

0.38 0.45 0.29

(21102110)

0.36 0.36 0.58

(21102110)

0.56 0.57 1.38

(21102110)

1.07

(42204220)

0.81 0.74 0.09

(21102110)

0.05 0.07 1.04

(21102110)

0.47 0.45

a Experimental data from ref 59. bExperimental data from ref 60 and 61. cExperimental data from ref 62. dExperimental data from ref 56 and 63. eExperimental data from ref 64. fFS-MR-CC from ref 9.

(21102110)

of electronic 2-spinors. The positronic solutions can be simply abandoned. Methods that allow such a separation of spectra are called two-component methods. Today, several such methods are available that allow for separating positive and negative spectra. In these methods, the separation of the atomic and molecular spectra is obtained with different accuracies.28,33 One of the methods, which allows for the generation of twocomponent solutions of arbitrarily high accuracy that are formally equivalent to solutions of the Dirac equation for the discrete electronic part on its eigenspectrum, has been proposed by Barysz et al.23−28 The method is formally of infinite order in the fine structure constant and has been called the infinite-order two-component (IOTC) theory. One of the most commonly used relativistic method in chemical applications is the Douglas−Kroll−Hess method in the DKH2 approximation, which gets the separation of positive and negative solutions accurate to the second order in the one electron potential V.34,35 In recent years, the method has been extended to higher orders (DKHn (n = 2, 3, 4, ···) approximations).36−38 It is an additional goal in this work to test the performance of the spin-free IOTC method and to compare its results with the DKHn approximations (n = 2, 3, 4,···, 10). The POL and POL.DK Gaussian basis sets have been used in the nonrelativistic and relativistic CASSCF/CASPT2 calculations. Gaussian basis sets [1310p4d/7s5p2d] and [21s17p11d9f/ 13s11p7d4f] have been used in the calculations for Si and Au, respectively.31 The basis set used differs a bit from those used earlier in our calculations,39 and they lead to slightly different

(21102110)

a

Experimental data from refs 56 and 57. bExperimental data from ref 58. cFS-MR-CC from ref 9.

needs to be replaced by its appropriate relativistic counterpart, which is the many-electron Dirac−Coulomb or Dirac−Breit Hamiltonian. The solutions of the Dirac equation have fourcomponents, and manipulating them in the many electron theory based on the Dirac−Coulomb Hamiltonian is quite cumbersome. Moreover, the use of the Dirac Hamiltonian requires considerably larger computational resources compared to the use of the Schrödinger Hamiltonian. The four-component relativistic Dirac wave function contains information regarding the positiveand negative-energy states of the system. In chemical applications, one is usually concerned with the electronic (or positive-energy) states only. Therefore, some reductions of the four-component wave function seem to be preferred. The exact separation of the positive and negative spectra would be equivalent to the transformation of the Dirac 4-spinors into either electronic or positronic 2-spinors. Alternatively, this means that the 4 × 4 Dirac Hamiltonian is block-diagonalized, i.e., brought into the form of the direct sum of the 2 × 2 matrix Hamiltonians with one of them corresponding to the electronic spectrum and the other referring solely to the positronic eigenvalues. Once this is achieved, most problems of the relativistic quantum chemistry can be formulated solely in terms 1622


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Table 16. Spectroscopic Constants for the Na2 Molecule in the ANO-RCC or POL Basis for the 3s + 4s Dissociation Limit in the CASSCF (31103110) Inactive Space state 23Σ+g

31Σ+g

21Σ+u

21Σ+u

a

Re (Å)

method IOTC CASSCF/CASPT2 NR CASSCF/CASPT2 FS-MR-CCd FS-MR-CCd expta IOTC CASSCF/CASPT2 NR CASSCF/CASPT2 FS-MR-CCd FS-MR-CCd exptb NR CASSCF/CASPT2 FS-MR-CCd exptc NR CASSCF/CASPT2 FS-MR-CCd exptc

3.35 3.58 3.35 3.62 3.53 4.33 3.63 4.30 3.69 3.57 3.71 3.82 3.69 6.77 6.78 6.74

ωe (cm−1)

De (eV)

ωexe (cm−1)

active space

1.78 0.25 0.46 0.33 0.46 0.71 0.76 1.57 0.88 0.86 0.72 5.90

(42204220) ANO-RCC (42204220) POL ANO-RCC POL

3s + 4s Dissociation Limit 0.89 148 0.74 122 0.94 149 0.72 120 0.77 127 0.93 77 0.70 105 0.94 79 0.72 97 0.75 112 0.44 114 0.37 107 0.37 106 0.47 51 0.47 51 0.48 53

0.14

(42204220) ANO-RCC (52215221) POL ANO-RCC POL (42204220) inner POL POL (42204220) outer POL POL

Experimental data from ref 65. bExperimental data from ref 66. cExperimental data from ref 67. dFS-MR-CC from ref 9.

Table 17. Adiabatic Excitation Energies for the Na2 Molecule in the ANO-RCC Basis Set state 13Σ+u

13Πu

11Σ+u

13Σ+g

11Πu

21Σ+g

11Πg

method

ANO-RCC (eV)

3s + 3s Dissociation Limit IOTC CASSCF/CASPT2 0.72 FS-MR-CCb 0.75 expta 0.73 3s + 3p Dissociation Limit IOTC CASSCF/CASPT2 1.68 FS-MR-CCb 1.66 expta 1.68 IOTC CASSCF/CASPT2 1.81 FS-MR-CCb 1.81 expta 1.82 IOTC CASSCF/CASPT2 2.27 FS-MR-CCb 2.26 expta 2.26 IOTC CASSCF/CASPT2 2.56 FS-MR-CCb 2.52 expta 2.52 IOTC CASSCF/CASPT2 2.40 FS-MR-CCb 2.41 expta 2.40 IOTC CASSCF/CASPT2 2.69 FS-MR-CCb 2.70 expta 2.70

Table 18. Adiabatic Excitation Energies for the Na2 Molecule in the POL Basis Set for the 3s + 4s Dissociation Limit

active space

state

(21102110)

23Σ+g

31Σ+g (21102110) 21Σ+u (21102110) 21Σ+u (21102110)

method

POL (eV)

3s + 4s Dissociation Limit CASSCF/CASPT2 3.11 FS-MR-CCb 3.13 expta 3.13 CASSCF/CASPT2 3.14 FS-MR-CCb 3.14 expta 3.18 CASSCF/CASPT2 3.43 FS-MR-CCb 3.49 expta 3.53 CASSCF/CASPT2 3.37 FS-MR-CCb 3.38 expta 3.46


(42204220)

(42204220) inner

(42204220) outer

a

Experimental data from ref 58, 59, 61, 63, 64, and 68. bFS-MR-CC from ref 9.

(21102110)

molecule well, and I expect the same accuracy for the other excited states. The IOTC values also agree very well with the calculated DKHn parameters in all DKHn approximations (n = 2, 3, 4, ···, 10). It can be seen that the degree of approximation used in the DKHn method practically does not change the calculated values of the parameters. Analyzing the results of the calculations, one can observe quite a large role of the relativistic effect. For example, the calculated IOTC relativistic dissociation energies for the ground 2Πr state is 3.21 eV, and the corresponding nonrelativistic value is only 1.53 eV. Similarly, the relativistic IOTC dissociation energy for the 2Δ state is 2.75 eV, and the nonrelativistic energy is 0.86 eV. Note, however, that this state does not dissociate into the lowest atomic dissociation limit but to the Au0(2S) + Si0(1D) dissociation limit. To conclude, the relativistic theory is necessary to properly described the basic spectroscopic constants Re, De, and ωe of heavy molecules in their ground and excited states. Finally, I should also stress that the present relativistic IOTC approach is based on the one-component (spin-free) approximation. Thus, it neglects all spin−orbit coupling contributions.

(21102110)

(21102110)

a

Experimental data from ref 58, 59, 61, 63, 64, and 68. bFS-MR-CC from ref 9.

values of the total energy. All calculations have been carried out in C2v symmetry. The partition of the orbital space used in CASSCF calculation is (0.0.0.0/20.10.10.4/3.1.1.0; 5el]. In the CASPT2 method the partition is (17.8.8.3/3.2.2.1/3.1.1.0; 5el). The theoretical results and available experimental data for the ground 2Πr state and three lowest excited states 4Σ−, 2Σ−, and 2Δ of the SiAu molecule are presented in Table 20. On the basis of the data in Table 20, it can be seen that the IOTC CASSCF/CASPT2 method reproduce the experimental spectroscopic parameters of the ground state of the SiAu 1623


Article


perturbative DKHn results (n = 2, 3, 4, ···, 12) converge to the IOTC result, but they go a little below the IOTC value; the difference is approximately 0.03 au. It should be noted, however, that the parametrizations implemented in the DKHn approximation do not affect the Hamiltonian only up to the fourth order. Therefore, as long as one runs calculations with the DKHn Hamiltonian below the fifth order, any parametrization can be used as they would all yield the same results. A higher order DKHn Hamiltonian depends slightly on the chosen parametrization of the unitary transformations applied to decouple the Dirac Hamiltonian.22

Table 19. Energy in the Dissociation Limit for the Na2 Molecules in the IOTC CASSCF/CASPT2 Calculations in the ANO-RCC Basis for the 3s + 3s and 3s + 3p Dissociation Limit and in the CASSCF/CASPT2 Calculations and the POL Basis for the 3s + 4s Dissociation Limit molecular symmetry X1Σ+g 13Σ+u 2s + 2s Li atom 13Πu 11Σ+u 13Σ+g 11Πu 21Σ+g 11Πg 2s + 2p Li atom 23Σ+g 31Σ+g 21Σ+u 2s + 3s Li atom a

energy (au)

ΔEa

3s + 3s Dissociation Limit −324.596803 0.000055 −324.596781 0.000033 −324.596748 3s + 3p Dissociation Limit −324.520015 0.000084 −324.519937 0.000006 −324.519569 0.000362 −324.519999 0.000068 −324.519891 0.000040 −324.520042 0.000111 −324.519931 3s + 4s Dissociation Limit −323.726755 0.000036 −323.726752 0.000039 −323.726770 0.000021 −323.726791

active space (21102110) (21102110)

(21102110) (21102110) (21102110) (21102110) (21102110) (21102110)

VI. SUMMARY In this study, it has been shown that the CASSCF/CASPT2 method allows me to describe the low and highly excited states with very good accuracy, practically as good as the results obtained recently by the multireference FS-MR-CC coupled cluster method. The final results of both methods depend strongly on the basis set and the selected active space. Rather small POL basis sets seem to be very good for the calculations of a wide range of states, including the highly excited states. Though the POL basis has been designed to calculate electric properties of the molecules, it has already been noticed quite some time ago that it can properly describe the properties of the ground and excited states of the molecules. My work only confirms the validity of this observation. The relativistic contracted ANORCC basis, though designed for ground and excited states, works good but often fails for the highest excites states. The basis appears to be too small for such demanding calculations. The relativistic effects for the molecules composed of heavy elements is a must. The strong relativistic effects in the gold atom are a very well-known fact, and the same has been expected for the AuSi molecule. My results confirm this expectation and show that the relativistic effects are important not only for the ground state of the AuSi molecule but for the excited states as well. The relativistic infinite-order two-component (IOTC) method works very well for all studied states. Although the calculated IOTC

(42204220) (42204220) (42204220)

ΔE = E(Na2) − E(Na) − E(Na) (atoms in the dissociation limit).

Because the open shell structure of the ground and excited states is mostly associated with the open shell structure of Si, the spin− orbit coupling effects should not significantly contribute to the calculated molecular properties, in particular, to the theoretically predicted ordering of different low-lying excited states. To plot the potential energy curve, one needs the total energy values. The relativistic IOTC and DKHn values of the total energies of the ground state and 2Σ− excited state in their equilibrium distances are shown in Table 21. According to my calculations, the total energy obtained in the Douglas−Kroll− Hess method strongly depends on the approximations used. The

Table 20. Spectroscopic Constants and Adiabatic Excitation Energies (ΔE) for the SiAu Molecule in the POL.DK and POL Basis for the IOTC, DKHn, and NR CASSCF/CASPT2 Calculations state

Re (Å)

method 0 2

ground state 2Πr

4 −

Σ state

2 −

Σ state

Δ state

2

a

De (eV)

ωe (cm−1)

ΔE (eV)

395 394 394 393 394 395 255 400 356 250 240 239 240 239 145

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.77 0.48 2.65 2.65 2.67 2.65, 2.66, 2.66, 2.65, 2.65 1.27

437 235

1.70 1.60

0 3

Au ( S) + Si ( P) Dissociation Limit IOTC CASSCF/CASPT2 2.25 3.21 DKH2 2.25 3.22 DKH3 2.25 3.22 DKH4 2.25 3.21 DKH5 2.25 3.22 DKH6, DK7, DK8, DK10 2.25 3.21 NR CASSCF/CASPT2 2.51 1.53 expta 2.26 3.3 IOTC CASSCF/CASPT2 2.26 1.48 NR CASSCF/CASPT2 2.51 1.08 IOTC CASSCF/CASPT2 2.40 0.55 DKH2 2.40 0.57 DKH3 2.40 0.58 DKH4, DKH5, DKH6, DK7, DK8, DK10 2.40 0.58 NR CASSCF/CASPT2 2.66 0.29 Au0(2S) + Si0(1D) Dissociation Limit IOTC CASSCF/CASPT2 2.22 2.75 NR CASSC/CASPT2 2.50 0.86

Experimental data from refs 69−71. 1624


Article


Table 21. Comparison of the DKHn (n = 2, 4, 6, 8) and the IOTC Energies (in a.u.) at the Equilibrium Distance Re for the 2Πr and 2 − Σ States of the SiAu Molecule state

method

energy (au)

state

Πr

NR DKH2 DKH3 DKH4 DKH5 DKH6 DKH7 DKH8 DKH10 DKH12 IOTC

−18154.3653488760 −19282.3655269116 −19302.1657247104 −19299.4303058818 −19300.6716168481 −19300.1562272142 −19300.3077075913 −19300.2567199547 −19300.2681874299 −19300.2699035495 −19300.2383780927

2 −

2

Σ

AUTHOR INFORMATION

Notes

The author declares no competing financial interest.

■

energy (au)

NR DK2 DK3 DK4 DK5 DK6 DK7 DK8 DK10 DK12 IOTC

−18154.3186431234 −19282.2681277569 −19302.0676214906 −19299.3328482320 −19300.5739025659 −19300.0587442057 −19300.2099968179 −19300.1592332884 −19300.1707040652 −19300.1724170129 −19300.1408893354

(21) Iliaś, M.; Kellö, V.; Urban, M. Acta Phys. Slovaca 2010, 60, 259− 391. (22) Molcas7.3, a program written by: Karlstrom, G.; Lindh, R.; Malmqvist, P.-A.; Roos, B. O.; Ryde, U.; Veryazov, V.; Widmark, P.-O.; Cossi, M.; Schimmelpfennig, B.; Neogrady, P.; Seijo, L. Comput. Mater. Sci. 2003, 28, 222. (23) Barysz, M.; Sadlej, A. J. J. Chem. Phys. 2002, 116, 2696−2704. (24) Kȩdziera, D.; Barysz, M. J. Chem. Phys. 2004, 121, 6719−6727. (25) Kȩdziera, D.; Barysz, M.; Sadlej, A. J. Struct. Chem. 2004, 15, 369− 377. (26) Barysz, M.; Leszczyński, J. J. Chem. Phys. 2007, 126, 154106−1−6. (27) Barysz, M.; Mentel, Ł.; Leszczyński, J. J. Chem. Phys. 2009, 130, 164114−1−8. (28) Barysz, M. In Relativistic Methods for Chemists; Barysz, M., Ishikawa, Y., Eds.; Springer: New York, USA, 2010; pp 165−190. (29) Barysz, M.; Syrocki, Ł. Mol. Phys. 2014, 112, 583−591. (30) The IOTC method was implemented in the version of the Molcas 7.3 system of programs by Slovakia Group of Quantum Chemistry (Comenius University, Bratislava, Slovakia, 2010). The corresponding patches for Molcas 7.3 release can be obtained directly from M. Barysz (e-mail: [email protected]). (31) Cernusak, I.; Kellö, V.; Sadlej, A. J. Collect. Czech. Chem. Commun. 2003, 68, 211−239. (32) Roos, B. O.; Veryazov, V.; Widmark, P.-O. Theor. Chem. Acc. 2004, 111, 345−352. (33) Reiher, M.; Wolf, A. Relativistic Quantum Chemistry; Wiley-VCH Verlag GmbH and Co. KGAa: Weinheim, 2009; pp 413−500. (34) Hess, B. A. Phys. Rev. A: At., Mol., Opt. Phys. 1985, 32, 756−763. (35) Hess, B. A. Phys. Rev. A: At., Mol., Opt. Phys. 1986, 33, 3742−3748. (36) Wolf, A. J.; Reiher, M.; Hess, B. A. J. Chem. Phys. 2002, 117, 9215− 9226. (37) Reiher, M.; Wolf, A. J. J. Chem. Phys. 2004, 121, 2037−2047. (38) Reiher, M.; Wolf, A. J. J. Chem. Phys. 2004, 121, 10945−10956. (39) Turski, P.; Barysz, M. J. Chem. Phys. 2000, 113, 4654−4661. (40) Hessel, M. M.; Vidal, C. R. J. Chem. Phys. 1979, 70, 4439−4459. (41) Barakat, B.; Bacis, R.; Carrot, F.; Churassy, S.; Crozet, P.; Martin, F.; Verges, J. Chem. Phys. 1986, 102, 215−227. (42) Xie, X.; Field, R. W. J. Chem. Phys. 1985, 83, 6193−6196. (43) Linton, C.; Murphy, T. L.; Martin, F.; Bacis, R.; Verges, J. J. Chem. Phys. 1989, 91, 6036−6041. (44) Engelke, F.; Hage, H. Chem. Phys. Lett. 1983, 103, 98−102. (45) Kusch, P.; Hessel, M. M. J. Chem. Phys. 1977, 67, 586−589. (46) Barakat, B.; Bacis, R.; Churassy, S.; Field, R. W.; Ho, J.; Linton, C.; McDonald, S. M.; Martin, F.; Verges, J. J. Mol. Spectrosc. 1986, 116, 271− 285. (47) Miller, D. A.; Gold, L. P.; Tripodi, P. D.; Bernheim, R. A. J. Chem. Phys. 1990, 92, 5822−5825. (48) He, C.; Gold, L. P.; Bernheim, R. A. J. Chem. Phys. 1991, 95, 7947−7951. (49) Lazarov, G.; Lyyra, A. M.; Li, L. J. Mol. Spectrosc. 2001, 205, 73− 85.

spectroscopic parameters are of the same quality I get from the Douglas−Kroll−Hess (DKHn) calculations (for all n = 2, 3, 4, ··· orders of approximations), the total IOTC energy curves are much more accurate than the energies obtained by the DKHn method. For the IOTC accuracy to be obtained, high order DKHn (n = 10 or 12) is necessary.

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method

REFERENCES

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CASPT2 and FS-MR-CC

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