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Equation-of-motion coupled cluster method for ionized states with partial inclusion of connected triples: Assessment of the accuracy in regular and explicitly-correlated approaches
Chemical Physics Letters 610–611 (2014) 173–178
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Equation-of-motion coupled cluster method for ionized states with partial inclusion of connected triples: Assessment of the accuracy in regular and explicitly-correlated approaches Denis Bokhan a,∗ , Dmitrii N. Trubnikov a , Monika Musiał b , Rodney J. Bartlett c a Laboratory of Molecular Beams, Physical Chemistry Division, Department of Chemistry, Moscow Lomonosov State University, Moscow 119991, Russian Federation b Institute of Chemistry, University of Silesia, Katowice, Poland c Quantum Theory Project, University of Florida, Gainesville, FL, United States
a r t i c l e
i n f o
Article history: Received 27 March 2014 In final form 1 July 2014 Available online 8 July 2014
a b s t r a c t Equation-of-motion coupled cluster method for ionized states with partial inclusion of connected triples is implemented within both regular and explicitly-correlated approaches. The computational scaling of proposed scheme is N6 , so the IP-EOM part is not more expensive than the underlying neutral-state CCSD calculation. Numerical results for the set of molecules and their ionized states are in good agreement with highly-accurate IP-EOM-CCSDT results. Comparison of predicted and experimental results for target ionization potentials shows an agreement between two sets with average deviation of ∼0.2 eV. Better agreement with IP-EOM-CCSDT results indicate that the most significant discrepancies may be related to the not enough accurate values of vertical ionization potentials, restored from experimental data. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The EOM-CC methodology is a convenient tool for the calculation of differential energies, such as ionization potentials (IP-EOM-CC) [1], electron affinities (EA-EOM-CC) [2], excitation energies (EE-EOM-CC) [3] double ionization potential (DIP) [4], and double electron attached (DEA) [5] states, and some other properties [6]. The key step of any EOM-CC calculation is the diag-
TH eT onalization of a non-Hermitian effective Hamiltonian H = e− based upon the ground or reference state amplitudes, T, in the corresponding sector of Fock space. The N-h and N+p spaces of H are used to obtain IPs and EAs, while the N-electron sector gives excitation energies. The IP-EOM-CC and EA-EOM-CC are directly related to the 1h and 1p sectors of the Fock-space coupled-cluster approach but provide a far better computational procedure. Thus they exhibit exact size-extensivity [7–9]. The related similarlytransformed EOM-CC (STEOM-CC) method has been developed by Nooijen and Bartlett [10] as a computationally less demanding alternative to EE-EOM-CC.
∗ Corresponding author. E-mail address: [email protected] (D. Bokhan). http://dx.doi.org/10.1016/j.cplett.2014.07.001 0009-2614/© 2014 Elsevier B.V. All rights reserved.
Nowadays, the EOM-CC algorithm [11,12], based on coupledcluster singles and doubles (CCSD) has become a standard tool in quantum chemical calculations and can be routinely used for the treatment of molecular systems. The calculation of the principal IPs with the IP-EOM-CCSD method can provide an accuracy of a few tenths of electron-volt (eV) with respect to experimental data if reasonably augmented Gaussian bases are used [6]. The IPEOM-CCSDT method, developed by Musiał et al. [13] significantly improves the accuracy, but the computational cost of this method is proportional to N8 , and, thus, can be applied only for small systems. On the other hand, the accuracy of the ionization potentials obtained is known to be quite dependent upon the chosen basis set. In order to circumvent this problem, the explicitly-correlated IPEOM-CCSD(F12) was developed [14]. Results, reported in Ref. [14] indicate significant acceleration of convergence of target ionization potentials with the maximal angular momentum of the basis set, but comparison with experimental results shows an increase in the error with the number of electrons in the system which should not happen for a size-extensive method. Such inaccuracy can potentially be eliminated by the inclusion of triple excitations into the IP-EOM-CCSD(F12) scheme, which would have a computational cost proportional to N7 . In this Letter we explore an even more approximate method, IP-EOM(R3)-CCSD(F12) which scales
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like N6 and includes both short-range Slater geminals and triple excitations to offer a balanced treatment of differential correlation energies. Following a discussion of the theory, a numerical test for a set of molecules and assessment is presented in the subsequent sections. 2. Theory 2.1. General considerations Within the coupled-cluster (CC) theory the ground-state wave function of a neutral system is written in the form T1 + T2 +···) 0 = e( ˚0 ,
(1)
Tn are regular cluster operators and ˚0 – any single deterwhere minant reference, but frequently the ground-state Hartree–Fock determinant. The details of CC theory with the corresponding working equations have been presented in numerous articles and textbooks. In this work the coupled cluster singles and doubles (CCSD) neutral-state wave function will be used: T1 + T2 ) 0 (CCSD) = e( ˚0 .
(2)
The linearly approximated explicitly-correlated extension of CCSD, known as CCSD(F12) [15] includes an additional operator, T2 which takes care of short-range correlation effects: 0 (CCSD(F12)) = e
2 ˚ . 0
(T1 +T2 +T )
(3)
T2 operator has the form: The 1 ij T2 = tkl ( ˛ˇ|f12 |kl E˛i Eˇj − ab|f12 |kl Eai Ebj ). 2
ijkl
˛ˇ
2.2. Working equations (4)
ab
Here Epq denote unitary group generators, + Epq = a+ p↑ aq↑ + ap↓ aq↓
(5)
and f12 are Slater-type geminals [16]: f12
1 = − exp(−r12 ).
(6)
The symbols i, j, . . . , a, b, . . ., ˛, ˇ, . . ., p, q, . . . are used respectively for occupied, virtuals, virtuals from the complete basis and all possible spin–orbitals. In our approach the IP-EOM-CC wave function of the target ionized state assumes the following form: ion = R0 (CCSD/CCSD(F12))
(7)
R=( R1 + R2 + R3 ), and where
R1 =
ri ai
(8)
i
R2 =
†
rija {aa ai aj }
(9)
i,j,a
R3 =
† †
ab rijk {aa ab ai aj ak }
(10)
i,j,k,a,b
and 0 (CCSD/CCSD(F12)) stands for either the regular or the explicitly-correlated CCSD solution. Introducing the effective Hamiltonian T T + [H , T3 ] H = e− He =H
(11)
= e(−T1 −T2 ) He(T1 +T2 ) H
(12)
the IP-EOM-CC equations can be written as
, [H R]|˚0 = ωk R|˚0 ,
where ωn is the ionization potential. The explicitly-correlated version of Eq. (13) has the same form as regular IP-EOM, but will be augmented by terms originating from elements of H geminals; detailed description is given in Ref. [14]. There is no direct contribution of (F12) to the R-equations for the IP problem. Traditionally, IP-EOM methods are formulated in such a way that the excitation levels of the T and R operators are the same. The IP-EOM-CCSD and IP-EOM-CCSDT are examples of that. However, In our IP-EOM(R3)-CCSD approach the T operator is taken from CCSD, while the R operator is truncated at the level of triples. does not satisfy the coupled cluster T3 equation, Even though H the structure of the EOM-CC equations is preserved for the case of ionization potentials and electron affinities, while for excitation energies an extra term will appear due to the presence of the r0 term. The main idea of an IP-EOM approach is to describe ionized states in terms of neutral state orbitals and cluster amplitudes and each term of Eq. (7) has its purpose. The R1 operator creates ionized states by annihilation of one electron (one hole, 1h). The R2 operator (2h1p) provides partial orbital relaxation in the ionized state, while for more complete inclusion of relaxation effects higher-order terms would be necessary. The double excitations of R3 (3h2p) take into account correlation effects in the ionized state, thus, helping to provide a proper description of differential correlation effects.
(13)
Eq. (13) can be written in terms of projections:
, ˚i |[H R]|˚0 = ωk ˚i | R|˚0
(14)
, ˚aij |[H R]|˚0 = ωk ˚aij | R|˚0
(15)
, ˚ab |[H R]|˚0 = ωk ˚ab | R|˚0 ijk ijk
(16)
of which those due to R3 are of interest, the rest simply being the R3 specific terms are IP-EOM-CCSD part of the equations. These
, ˚i |[H R3 ]|˚0 = ωk ˚i | R1 |˚0 , ˚aij |[H R3 ]|˚0 = ωk ˚aij | R2 |˚0 , ( ˚ab |[H R1 + R2 + R3 )]|˚0 = ωk ˚ab | R |˚0 ijk ijk 3 In our treatment of the R3 operator we choose to exclude any terms that scale worse than N6 to be consistent with the computational dependence of the CCSD reference state. We also eliminate the R3 contributions to R2 and the R2 contribution to R3 as a part of this approximation. So far, in the framework of our approach only R3 and R1 related terms are coupled directly, i.e. equations for R1 contain contribution from R3 and equations for R3 contain only contribution from R1 . Such coupling provides account of influence of differential correlation effects on ionization amplitudes in R1 operator, and, thus more accurate vertical ionization potentials are expected. Decoupling of R3 and R2 preserves N6 computational scaling, but description of interference of relaxation and differential correlation effects cannot be done in such imbalanced scheme. As a consequence, results for ionized states with significant relaxation, like inner valence or core states may suffer from errors. Nevertheless, the IP-EOM(R3)-CCSD should be less imbalanced and more accurate then the traditional IP-EOM-CCSD method without any R3 terms. inclusion of
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is performed using an extended Davidson algorithm matrix of H with the simultaneous search of left and right eigenvectors. The explicitly-correlated CCSD(F12) method is implemented within the B-approximation according to Ref. [18]. Related Slater and Yukawa integrals are implemented using the Rys quadrature technique [19]. Many-electron integrals are calculated with numerical quadratures, using Becke’s fuzzy cell method [20]. The atomic grids used for the evaluation of many-electron integrals are constructed using 50-point radial grids and for angular integrations, 194-point Lebedev–Laikov grids [21].
In terms of diagrams, (14)–(16) become,
3. Results and discussion
where wiggly lines denote the corresponding elements of similarly . In the first diagram the wavy line and transformed hamiltonian H the usual dashed line are the same. The last term on the right-hand site of R3 consists of two terms:
(17) Diagrams in the right-hand side of Eq. (17) can be represented in terms of intermediates that makes possible the elimination of one occupied index, so its computational cost becomes proportional to N6 . Thus, this procedure is no more expensive then the underlying neutral state CCSD calculation. 2.3. Details of implementation The IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) are implemented in the ACES III massively parallel quantum chemistry software package [17]. The diagonalization of the non-Hermitian
In order to test the accuracy of the developed IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) methods, vertical ionization potentials of 11 molecules are calculated using aug-cc-pCVXZ bases, where X = D and T [22–24]. All the calculations are performed using experimental equilibrium geometries, available online [25]. The value of the exponential parameter in Eq. (6) is equal to 1.5. Results for different methods are presented in Tables 1 and 2. In all methods, regular or explicitly-correlated, the inclusion of the R3 operator makes the values of the IPs lower. A similar tendency is observed for the full IP-EOM-CCSDT results compared to CCSD. For 10-electron systems, like Ne, HF, H2 O and NH3 , the average difference between the IP-EOM and IP-EOM(R3) results is less than 0.1 eV for both basis sets. In the case of 14-electrons, like in N2 , CO, HCN and C2 H2 that difference increases to 0.17 and 0.19 eV for double and triple bases, respectively. The difference grows with the correlation energy of the ground state (and, thus, the number of electrons). This fact emphasizes the role of R3 in the description of the differential correlation energy. Comparison with the available experimental results shows that in the case of the triple- basis the inclusion of R3 tends to improve the explicitly-correlated results for the ionization potentials. The statistical measures in Table 3 indicates, however, that all of the methods exhibit a similar level of accuracy with a standard deviation of 0.26–0.28 eV and maximal absolute error of 1 eV. Indeed, even in the relatively simple case of the HF molecule, the deviation between IP-EOM(R3)-CCSD(F12) and experiment for the first root is equal to 0.16 eV Even worse results are observed for some roots in H2 O, HCN, C2 H2 , C2 H4 and CH2 O. In order to find the origin of the large discrepancies between experiment and theory, IP-EOM-CCSDT results, calculated at the complete basis set (CBS) limit are also presented in Tables 1 and 2. The CBS extrapolation of the IP-EOM-CCSDT results is performed using the two-point formula [26] for triple and quadruple- bases. It should be noted that only this result contains the ground state . Results of T3 contributions to fully define H. All the rest use H IP-EOM(R3)-CCSD(F12) with both double and triple- basis are close to the IP-EOM-CCSDT values, and the corresponding statistical measures are presented in Table 4. Among all considered methods IP-EOM(R3)-CCSD(F12) seems to be the most accurate to be used for an estimation of vertical ionization potentials even at a double- level. The largest deviation of 0.26 eV is observed for the 1b2 state of formaldehyde and should be related to the approximations made in the equations for the R3 terms, though the ground state T3 in the full triples method might have a small role. Note that IP-EOM(R3)-CCSD results somewhat erratic (with respect to IP-EOM-CCSDT/CBS) even in the case of triple- basis. Comparison with the corresponding IP-EOM-CCSDT results in fourth column of Table 2 indicate that except last roots of Ne, F2 , CH2 O and C2 H4 difference with IP-EOM(R3)-CCSD values is quite modest. In fact, observed errors are not surprising, since all those last roots are correspond to inner valence ionization, where relaxation effects are significant. For the case of explicitly-correlated approach situation is similar: difference between IP-EOM(R3)-CCSD(F12) and
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Table 1 Vertical ionization potentials (in eV) for the set of molecules, calculated using AUG-CC-PCVDZ basis set. IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
2p 2s
21.18 48.38
21.22 48.48
21.80 48.93
21.81 49.00
21.55 48.22
21.56 48.42
1 3
15.87 19.91
15.85 19.88
16.36 20.31
16.33 20.27
16.22 20.12
16.05 20.00
15.45 17.12 18.75
15.26 16.83 18.60
15.84 17.46 19.07
15.63 17.16 18.91
15.63 17.09 18.82
15.60 16.98 18.78
14.01 16.93 19.70
13.83 16.75 19.65
14.29 17.30 20.09
14.11 17.12 20.03
14.04 17.13 19.70
14.01 16.91 19.72
15.42 18.79 21.18
15.31 18.72 20.94
16.01 19.35 21.61
15.88 19.25 21.36
15.85 19.02 21.23
15.83 18.8 21.1
13.78 13.79 20.50
13.54 13.65
14.05 14.12 20.74
13.81 13.98 20.61
13.86 13.99 20.54
13.61 14.01 20.1
12.41 14.66 18.98
12.34 14.60 18.93
12.78 14.98 19.26
12.72 14.93 19.21
12.74 14.90 19.14
12.62 14.74 18.51
10.63 14.53 16.00 17.29
10.56 14.35 15.92 17.19
10.97 14.81 16.36 17.56
10.89 14.63 16.28 17.46
10.94 14.70 16.15 17.19
10.88 14.5 16.0 16.6
10.55 13.02 14.74 16.19 19.47
10.39 12.95 14.63 16.12 19.41
10.73 13.20 15.00 16.38 19.67
10.57 13.14 14.89 16.32 19.61
10.70 13.15 14.83 16.18 19.35
10.95 12.95 14.88 16.34 19.40
10.69 16.5
10.60 16.4
10.96 16.7
10.87 16.6
10.97 16.6
10.85 16.5
11.47 17.12 19.02
11.27 17.00 18.93
11.68 17.38 19.23
11.49 17.27 19.14
11.54 17.18 19.06
11.40 16.30 18.39
Ne
HF
N2 3 g 1u 2 u CO 5 1 4 F2 1g 1u 3 g HCN 1 3 2 H2 O 1b1 3a1 1b2 CH2 O 2b2 1b1 5a1 1b2 C2 H4 1b2u 1b2g 3ag 1b3u 2b1u NH3 3a1 1e C2 H2 1g 3 u 3 u
Table 2 Vertical ionization potentials (in eV) for the set of molecules, calculated using AUG-CC-PCVTZ basis set.
Ne 2p 2s HF 1 3 N2 3 g 1u 2 u CO 5 1 4 F2 1g 1u 3 g HCN 1 3 2
IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSDT
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
21.38 48.42
21.36 48.47
21.41 48.19
21.60 48.65
21.57 48.69
21.55 48.22
21.56 48.42
16.08 20.01
16.03 19.95
16.10 20.03
16.26 20.16
16.21 20.10
16.22 20.12
16.05 20.00
15.70 17.28 18.95
15.48 16.97 18.79
15.54 16.99 18.74
15.84 17.41 19.07
15.62 17.10 18.91
15.63 17.09 18.82
15.60 16.98 18.78
14.22 17.11 19.84
14.03 16.91 19.76
13.95 17.03 19.60
14.33 17.26 20.00
14.14 17.05 19.92
14.04 17.13 19.70
14.01 16.91 19.72
15.66 19.02 21.22
15.51 18.91 20.94
15.69 18.89 21.11
15.88 19.22 21.38
15.73 19.11 21.10
15.85 19.02 21.23
15.83 18.8 21.1
13.95 14.01
13.69 13.85 20.52
13.76 13.89 20.44
14.06 14.14
13.80 13.98 20.62
13.86 13.99 20.54
13.61 14.01 20.1
D. Bokhan et al. / Chemical Physics Letters 610–611 (2014) 173–178
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Table 2 (Continued)
H2 O 1b1 3a1 1b2 CH2 O 2b2 1b1 5a1 1b2 C2 H4 1b2u 1b2g 3ag 1b3u 2b1u NH3 3a1 1e C2 H2 1g 3 u 3 u
IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSDT
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
12.64 14.81 19.07
12.54 14.74 19.00
12.64 14.83 19.07
12.78 14.94 19.18
12.69 14.87 19.10
12.74 14.90 19.14
12.62 14.74 18.51
10.87 14.67 16.16 17.46
10.78 14.47 16.06 17.34
10.84 14.61 16.06 17.10
11.00 14.78 16.30 17.56
10.90 14.58 16.20 17.45
10.94 14.70 16.15 17.19
10.88 14.5 16.0 16.6
10.69 13.15 14.89 16.31 19.60
10.52 13.07 14.76 16.23 19.53
10.65 13.10 14.76 16.13 19.28
10.76 13.23 15.00 16.39 19.68
10.59 13.15 14.87 16.31 19.60
10.70 13.15 14.83 16.18 19.35
10.95 12.95 14.88 16.34 19.40
10.89 16.6
10.78 16.5
10.88 16.57
10.99 16.7
10.88 16.6
10.97 16.6
10.85 16.5
11.60 17.25 19.15
11.39 17.13 19.05
11.47 17.16 18.99
11.69 17.38 19.23
11.48 17.23 19.13
11.54 17.18 19.06
11.40 16.30 18.39
Table 3 ), mean absolute error Statistical measurements of the errors in ionization potentials with respect to experimental data: mean error (), maximum absolute error (max abs (abs ) and standard deviation (std ). IP-EOM-CCSD AUG-CC-PCVDZ AUG-CC-PCVTZ max abs AUG-CC-PCVDZ AUG-CC-PCVTZ
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS 0.15
0.02 0.17
−0.03 0.05
0.35 0.30
0.25 0.18
0.82 0.95
0.70 0.83
1.08 1.08
0.97 0.93
abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.20 0.21
0.23 0.17
0.37 0.31
0.28 0.21
std AUG-CC-PCVDZ AUG-CC-PCVTZ
0.29 0.27
0.29 0.28
0.27 0.26
0.29 0.27
0.67
0.20
0.25
IP-EOM-CCSDT/CBS values grows when more inner ionization is considered. This also suggest that orbital relaxation, which is more important in core states than in valence ones, is not being handled well enough as one would expect, lacking an exponential operator analogous to exp( T1 ) for R1 in the target state [27], and coupling between R3 and R2 equations.
The IP-EOM-CCSDT method is known to be very accurate and in its CBS limit results usually can be used as a reliable estimate of vertical ionization potentials. However, there are still some large deviations from experimental data even if a full-triples approach is used. Though there can still be questions about the role of R4 , which can be significant at times, see [6], equally likely errors derive
Table 4 ), mean absolute Statistical measurements of the errors in ionization potentials with respect to IP-EOM-CCSDT/CBS results: mean error (), maximum absolute error (max abs error (abs ), standard deviation (std ), and root mean square (RMS). IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
AUG-CC-PCVDZ AUG-CC-PCVTZ
0.14 −0.02
0.26 0.11
−0.19 −0.15
−0.07 −0.02
max abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.37 0.27
0.33 0.34
0.39 0.37
0.33 0.26
abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.16 0.09
0.25 0.14
0.19 0.15
0.10 0.08
0.14 0.12
0.14 0.12
0.11 0.10
0.11 0.10
0.20 0.12
0.29 0.16
0.22 0.18
0.13 0.10
std AUG-CC-PCVDZ AUG-CC-PCVTZ RMS AUG-CC-PCVDZ AUG-CC-PCVTZ
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from the difficult experimental extraction of vertical IPs. This issue has been previously studied for C2 H4 by Yau et al. [28] where it is suggested that theory might offer better values. Another possible source of error may be related to vibronic effects, causing difficulties in the experimental definition of vertical ionization. Surprisingly, for both double and triple- bases IP-EOM-CCSD are closer to IP-EOM-CCSDT/CBS results than the corresponding values of IP-EOM(R3)-CCSD. Doubtless, this is due to cancellation of errors and with greater basis completeness via (F12), the R operbalance may be compromised making the truncation of the ator more apparent. Indeed, comparing root mean square errors (RMS) in Table 4 for the double- basis one can see that IPEOM(R3)-CCSD(F12) method is significantly more accurate then all other methods. For triple- bases, where F12 correction is less important, IP-EOM(R3)-CCSD(F12) and IP-EOM-CCSD have similar RMS. One would hope that even more accurate results may be obtained if all diagrams contributing to the R3 operator are included in the corresponding EOM equation, but the computational cost of such a method would be proportional to N7 . 4. Conclusions The IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) methods are implemented and tested on a set of different systems and ionized states. In most cases IP-EOM(R3)-CCSD(F12) results even at double level are close to those of IP-EOM-CCSD/CBS. Taking into account the fact that the computational cost of IP-EOM(R3)-CCSD(F12) method is proportional to N6 , its results can be used as a reference for the interpretation of experimental data of photoelectron spectroscopy. For larger systems double- calculations are only feasible, and, since IP-EOM(R3)-CCSD(F12) appears to be the most accurate at this level among all considered methods, it can be recommended for practical calculations. Some large deviations from experimental results are likely related to non-vertical ionization of these small systems, encouraging the development and application of analytical gradients for IP-EOM to facilitate the determination of the experimentally observed adiabatic values when high accuracy is required. Presented study indicates that inclusion of R3 (even in its approximated form) and short-range correlation effects via
Slater geminals more important then the presence of T3 operator in neutral-state calculation. (The differences between using CCSDT-3 instead of CCSDT for the ground state T3 are negligible.) This conclusion will likely change for more difficult, multi-reference examples, but for the systems studied here, all differences fall within the statistical error bars. Acknowledgements We thank University of Florida High-Performance Computing Center (UF HPC) for providing computational facilities. This work has been supported partly by the US Air Force Office of Scientific Research. References [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]
R.J. Bartlett, J.F. Stanton, Rev. Comput. Chem. 5 (1994) 65. M. Nooijen, R.J. Bartlett, J. Chem. Phys. 102 (1995) 3629. J.F. Stanton, R.J. Bartlett, J. Chem. Phys. 98 (1993) 7029. R.J. Bartlett, Mol. Phys. 108 (2010) 2905. M. Musiał, S.A. Kucharski, R.J. Bartlett, J. Chem. Theory Comput. 7 (2011) 3088. R.J. Bartlett, M. Musiał, Rev. Mod. Phys. 79 (2007) 291. R.J. Bartlett, G.D. Purvis III, Int. J. Quantum Chem. 14 (1978) 561. D. Sinha, S.K. Mukhopadhyay, R. Chaudhuri, D. Mukherjee, Chem. Phys. Lett. 154 (1989) 544. D. Mukhopadhyay, S.K. Mukhopadhyay, R. Chaudhuri, D. Mukherjee, Theor. Chim. Acta 80 (1991) 441. M. Nooijen, R.J. Bartlett, J. Chem. Phys. 107 (1997) 6812. J. Geertsen, M. Rittby, R.J. Bartlett, Chem. Phys. Lett. 164 (1989) 57. D.C. Comeau, R.J. Bartlett, Chem. Phys. Lett. 207 (1993) 414. M. Musiał, S.A. Kucharski, R.J. Bartlett, J. Chem. Phys. 118 (2003) 1128. D. Bokhan, S. Ten-no, J. Chem. Phys. 132 (2010) 021101. D.P. Tew, W. Klopper, C. Hättig, Chem. Phys. Lett. 452 (2008) 326. S. Ten-no, Chem. Phys. Lett. 398 (2004) 56. V. Lotrich, N. Flocke, M. Ponton, A.D. Yau, S.A. Perera, E. Deumens, R.J. Bartlett, J. Chem. Phys. 128 (2008) 194104. D. Bokhan, S. Bernadotte, S. Ten-no, Chem. Phys. Lett 469 (2009) 214. T. Shiozaki, Chem. Phys. Lett. 479 (2009) 160. A.D. Becke, J. Chem. Phys. 88 (1987) 2547. V.I. Lebedev, D.N. Laikov, Doklady Math. 59 (1999) 477. T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6769. D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 103 (1995) 4572. http://webbook.nist.gov T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys. 106 (1997) 9639. T.J. Watson Jr., R.J. Bartlett, Chem. Phys. Lett. 555 (2013) 235. A.D. Yau, S.A. Perera, R.J. Bartlett, Mol. Phys. 100 (2002) 835.
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Equation-of-motion coupled cluster method for ionized states with partial inclusion of connected triples: Assessment of the accuracy in regular and explicitly-correlated approaches Denis Bokhan a,∗ , Dmitrii N. Trubnikov a , Monika Musiał b , Rodney J. Bartlett c a Laboratory of Molecular Beams, Physical Chemistry Division, Department of Chemistry, Moscow Lomonosov State University, Moscow 119991, Russian Federation b Institute of Chemistry, University of Silesia, Katowice, Poland c Quantum Theory Project, University of Florida, Gainesville, FL, United States
a r t i c l e
i n f o
Article history: Received 27 March 2014 In final form 1 July 2014 Available online 8 July 2014
a b s t r a c t Equation-of-motion coupled cluster method for ionized states with partial inclusion of connected triples is implemented within both regular and explicitly-correlated approaches. The computational scaling of proposed scheme is N6 , so the IP-EOM part is not more expensive than the underlying neutral-state CCSD calculation. Numerical results for the set of molecules and their ionized states are in good agreement with highly-accurate IP-EOM-CCSDT results. Comparison of predicted and experimental results for target ionization potentials shows an agreement between two sets with average deviation of ∼0.2 eV. Better agreement with IP-EOM-CCSDT results indicate that the most significant discrepancies may be related to the not enough accurate values of vertical ionization potentials, restored from experimental data. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The EOM-CC methodology is a convenient tool for the calculation of differential energies, such as ionization potentials (IP-EOM-CC) [1], electron affinities (EA-EOM-CC) [2], excitation energies (EE-EOM-CC) [3] double ionization potential (DIP) [4], and double electron attached (DEA) [5] states, and some other properties [6]. The key step of any EOM-CC calculation is the diag-
TH eT onalization of a non-Hermitian effective Hamiltonian H = e− based upon the ground or reference state amplitudes, T, in the corresponding sector of Fock space. The N-h and N+p spaces of H are used to obtain IPs and EAs, while the N-electron sector gives excitation energies. The IP-EOM-CC and EA-EOM-CC are directly related to the 1h and 1p sectors of the Fock-space coupled-cluster approach but provide a far better computational procedure. Thus they exhibit exact size-extensivity [7–9]. The related similarlytransformed EOM-CC (STEOM-CC) method has been developed by Nooijen and Bartlett [10] as a computationally less demanding alternative to EE-EOM-CC.
∗ Corresponding author. E-mail address: [email protected] (D. Bokhan). http://dx.doi.org/10.1016/j.cplett.2014.07.001 0009-2614/© 2014 Elsevier B.V. All rights reserved.
Nowadays, the EOM-CC algorithm [11,12], based on coupledcluster singles and doubles (CCSD) has become a standard tool in quantum chemical calculations and can be routinely used for the treatment of molecular systems. The calculation of the principal IPs with the IP-EOM-CCSD method can provide an accuracy of a few tenths of electron-volt (eV) with respect to experimental data if reasonably augmented Gaussian bases are used [6]. The IPEOM-CCSDT method, developed by Musiał et al. [13] significantly improves the accuracy, but the computational cost of this method is proportional to N8 , and, thus, can be applied only for small systems. On the other hand, the accuracy of the ionization potentials obtained is known to be quite dependent upon the chosen basis set. In order to circumvent this problem, the explicitly-correlated IPEOM-CCSD(F12) was developed [14]. Results, reported in Ref. [14] indicate significant acceleration of convergence of target ionization potentials with the maximal angular momentum of the basis set, but comparison with experimental results shows an increase in the error with the number of electrons in the system which should not happen for a size-extensive method. Such inaccuracy can potentially be eliminated by the inclusion of triple excitations into the IP-EOM-CCSD(F12) scheme, which would have a computational cost proportional to N7 . In this Letter we explore an even more approximate method, IP-EOM(R3)-CCSD(F12) which scales
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like N6 and includes both short-range Slater geminals and triple excitations to offer a balanced treatment of differential correlation energies. Following a discussion of the theory, a numerical test for a set of molecules and assessment is presented in the subsequent sections. 2. Theory 2.1. General considerations Within the coupled-cluster (CC) theory the ground-state wave function of a neutral system is written in the form T1 + T2 +···) 0 = e( ˚0 ,
(1)
Tn are regular cluster operators and ˚0 – any single deterwhere minant reference, but frequently the ground-state Hartree–Fock determinant. The details of CC theory with the corresponding working equations have been presented in numerous articles and textbooks. In this work the coupled cluster singles and doubles (CCSD) neutral-state wave function will be used: T1 + T2 ) 0 (CCSD) = e( ˚0 .
(2)
The linearly approximated explicitly-correlated extension of CCSD, known as CCSD(F12) [15] includes an additional operator, T2 which takes care of short-range correlation effects: 0 (CCSD(F12)) = e
2 ˚ . 0
(T1 +T2 +T )
(3)
T2 operator has the form: The 1 ij T2 = tkl ( ˛ˇ|f12 |kl E˛i Eˇj − ab|f12 |kl Eai Ebj ). 2
ijkl
˛ˇ
2.2. Working equations (4)
ab
Here Epq denote unitary group generators, + Epq = a+ p↑ aq↑ + ap↓ aq↓
(5)
and f12 are Slater-type geminals [16]: f12
1 = − exp(−r12 ).
(6)
The symbols i, j, . . . , a, b, . . ., ˛, ˇ, . . ., p, q, . . . are used respectively for occupied, virtuals, virtuals from the complete basis and all possible spin–orbitals. In our approach the IP-EOM-CC wave function of the target ionized state assumes the following form: ion = R0 (CCSD/CCSD(F12))
(7)
R=( R1 + R2 + R3 ), and where
R1 =
ri ai
(8)
i
R2 =
†
rija {aa ai aj }
(9)
i,j,a
R3 =
† †
ab rijk {aa ab ai aj ak }
(10)
i,j,k,a,b
and 0 (CCSD/CCSD(F12)) stands for either the regular or the explicitly-correlated CCSD solution. Introducing the effective Hamiltonian T T + [H , T3 ] H = e− He =H
(11)
= e(−T1 −T2 ) He(T1 +T2 ) H
(12)
the IP-EOM-CC equations can be written as
, [H R]|˚0 = ωk R|˚0 ,
where ωn is the ionization potential. The explicitly-correlated version of Eq. (13) has the same form as regular IP-EOM, but will be augmented by terms originating from elements of H geminals; detailed description is given in Ref. [14]. There is no direct contribution of (F12) to the R-equations for the IP problem. Traditionally, IP-EOM methods are formulated in such a way that the excitation levels of the T and R operators are the same. The IP-EOM-CCSD and IP-EOM-CCSDT are examples of that. However, In our IP-EOM(R3)-CCSD approach the T operator is taken from CCSD, while the R operator is truncated at the level of triples. does not satisfy the coupled cluster T3 equation, Even though H the structure of the EOM-CC equations is preserved for the case of ionization potentials and electron affinities, while for excitation energies an extra term will appear due to the presence of the r0 term. The main idea of an IP-EOM approach is to describe ionized states in terms of neutral state orbitals and cluster amplitudes and each term of Eq. (7) has its purpose. The R1 operator creates ionized states by annihilation of one electron (one hole, 1h). The R2 operator (2h1p) provides partial orbital relaxation in the ionized state, while for more complete inclusion of relaxation effects higher-order terms would be necessary. The double excitations of R3 (3h2p) take into account correlation effects in the ionized state, thus, helping to provide a proper description of differential correlation effects.
(13)
Eq. (13) can be written in terms of projections:
, ˚i |[H R]|˚0 = ωk ˚i | R|˚0
(14)
, ˚aij |[H R]|˚0 = ωk ˚aij | R|˚0
(15)
, ˚ab |[H R]|˚0 = ωk ˚ab | R|˚0 ijk ijk
(16)
of which those due to R3 are of interest, the rest simply being the R3 specific terms are IP-EOM-CCSD part of the equations. These
, ˚i |[H R3 ]|˚0 = ωk ˚i | R1 |˚0 , ˚aij |[H R3 ]|˚0 = ωk ˚aij | R2 |˚0 , ( ˚ab |[H R1 + R2 + R3 )]|˚0 = ωk ˚ab | R |˚0 ijk ijk 3 In our treatment of the R3 operator we choose to exclude any terms that scale worse than N6 to be consistent with the computational dependence of the CCSD reference state. We also eliminate the R3 contributions to R2 and the R2 contribution to R3 as a part of this approximation. So far, in the framework of our approach only R3 and R1 related terms are coupled directly, i.e. equations for R1 contain contribution from R3 and equations for R3 contain only contribution from R1 . Such coupling provides account of influence of differential correlation effects on ionization amplitudes in R1 operator, and, thus more accurate vertical ionization potentials are expected. Decoupling of R3 and R2 preserves N6 computational scaling, but description of interference of relaxation and differential correlation effects cannot be done in such imbalanced scheme. As a consequence, results for ionized states with significant relaxation, like inner valence or core states may suffer from errors. Nevertheless, the IP-EOM(R3)-CCSD should be less imbalanced and more accurate then the traditional IP-EOM-CCSD method without any R3 terms. inclusion of
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is performed using an extended Davidson algorithm matrix of H with the simultaneous search of left and right eigenvectors. The explicitly-correlated CCSD(F12) method is implemented within the B-approximation according to Ref. [18]. Related Slater and Yukawa integrals are implemented using the Rys quadrature technique [19]. Many-electron integrals are calculated with numerical quadratures, using Becke’s fuzzy cell method [20]. The atomic grids used for the evaluation of many-electron integrals are constructed using 50-point radial grids and for angular integrations, 194-point Lebedev–Laikov grids [21].
In terms of diagrams, (14)–(16) become,
3. Results and discussion
where wiggly lines denote the corresponding elements of similarly . In the first diagram the wavy line and transformed hamiltonian H the usual dashed line are the same. The last term on the right-hand site of R3 consists of two terms:
(17) Diagrams in the right-hand side of Eq. (17) can be represented in terms of intermediates that makes possible the elimination of one occupied index, so its computational cost becomes proportional to N6 . Thus, this procedure is no more expensive then the underlying neutral state CCSD calculation. 2.3. Details of implementation The IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) are implemented in the ACES III massively parallel quantum chemistry software package [17]. The diagonalization of the non-Hermitian
In order to test the accuracy of the developed IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) methods, vertical ionization potentials of 11 molecules are calculated using aug-cc-pCVXZ bases, where X = D and T [22–24]. All the calculations are performed using experimental equilibrium geometries, available online [25]. The value of the exponential parameter in Eq. (6) is equal to 1.5. Results for different methods are presented in Tables 1 and 2. In all methods, regular or explicitly-correlated, the inclusion of the R3 operator makes the values of the IPs lower. A similar tendency is observed for the full IP-EOM-CCSDT results compared to CCSD. For 10-electron systems, like Ne, HF, H2 O and NH3 , the average difference between the IP-EOM and IP-EOM(R3) results is less than 0.1 eV for both basis sets. In the case of 14-electrons, like in N2 , CO, HCN and C2 H2 that difference increases to 0.17 and 0.19 eV for double and triple bases, respectively. The difference grows with the correlation energy of the ground state (and, thus, the number of electrons). This fact emphasizes the role of R3 in the description of the differential correlation energy. Comparison with the available experimental results shows that in the case of the triple- basis the inclusion of R3 tends to improve the explicitly-correlated results for the ionization potentials. The statistical measures in Table 3 indicates, however, that all of the methods exhibit a similar level of accuracy with a standard deviation of 0.26–0.28 eV and maximal absolute error of 1 eV. Indeed, even in the relatively simple case of the HF molecule, the deviation between IP-EOM(R3)-CCSD(F12) and experiment for the first root is equal to 0.16 eV Even worse results are observed for some roots in H2 O, HCN, C2 H2 , C2 H4 and CH2 O. In order to find the origin of the large discrepancies between experiment and theory, IP-EOM-CCSDT results, calculated at the complete basis set (CBS) limit are also presented in Tables 1 and 2. The CBS extrapolation of the IP-EOM-CCSDT results is performed using the two-point formula [26] for triple and quadruple- bases. It should be noted that only this result contains the ground state . Results of T3 contributions to fully define H. All the rest use H IP-EOM(R3)-CCSD(F12) with both double and triple- basis are close to the IP-EOM-CCSDT values, and the corresponding statistical measures are presented in Table 4. Among all considered methods IP-EOM(R3)-CCSD(F12) seems to be the most accurate to be used for an estimation of vertical ionization potentials even at a double- level. The largest deviation of 0.26 eV is observed for the 1b2 state of formaldehyde and should be related to the approximations made in the equations for the R3 terms, though the ground state T3 in the full triples method might have a small role. Note that IP-EOM(R3)-CCSD results somewhat erratic (with respect to IP-EOM-CCSDT/CBS) even in the case of triple- basis. Comparison with the corresponding IP-EOM-CCSDT results in fourth column of Table 2 indicate that except last roots of Ne, F2 , CH2 O and C2 H4 difference with IP-EOM(R3)-CCSD values is quite modest. In fact, observed errors are not surprising, since all those last roots are correspond to inner valence ionization, where relaxation effects are significant. For the case of explicitly-correlated approach situation is similar: difference between IP-EOM(R3)-CCSD(F12) and
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Table 1 Vertical ionization potentials (in eV) for the set of molecules, calculated using AUG-CC-PCVDZ basis set. IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
2p 2s
21.18 48.38
21.22 48.48
21.80 48.93
21.81 49.00
21.55 48.22
21.56 48.42
1 3
15.87 19.91
15.85 19.88
16.36 20.31
16.33 20.27
16.22 20.12
16.05 20.00
15.45 17.12 18.75
15.26 16.83 18.60
15.84 17.46 19.07
15.63 17.16 18.91
15.63 17.09 18.82
15.60 16.98 18.78
14.01 16.93 19.70
13.83 16.75 19.65
14.29 17.30 20.09
14.11 17.12 20.03
14.04 17.13 19.70
14.01 16.91 19.72
15.42 18.79 21.18
15.31 18.72 20.94
16.01 19.35 21.61
15.88 19.25 21.36
15.85 19.02 21.23
15.83 18.8 21.1
13.78 13.79 20.50
13.54 13.65
14.05 14.12 20.74
13.81 13.98 20.61
13.86 13.99 20.54
13.61 14.01 20.1
12.41 14.66 18.98
12.34 14.60 18.93
12.78 14.98 19.26
12.72 14.93 19.21
12.74 14.90 19.14
12.62 14.74 18.51
10.63 14.53 16.00 17.29
10.56 14.35 15.92 17.19
10.97 14.81 16.36 17.56
10.89 14.63 16.28 17.46
10.94 14.70 16.15 17.19
10.88 14.5 16.0 16.6
10.55 13.02 14.74 16.19 19.47
10.39 12.95 14.63 16.12 19.41
10.73 13.20 15.00 16.38 19.67
10.57 13.14 14.89 16.32 19.61
10.70 13.15 14.83 16.18 19.35
10.95 12.95 14.88 16.34 19.40
10.69 16.5
10.60 16.4
10.96 16.7
10.87 16.6
10.97 16.6
10.85 16.5
11.47 17.12 19.02
11.27 17.00 18.93
11.68 17.38 19.23
11.49 17.27 19.14
11.54 17.18 19.06
11.40 16.30 18.39
Ne
HF
N2 3 g 1u 2 u CO 5 1 4 F2 1g 1u 3 g HCN 1 3 2 H2 O 1b1 3a1 1b2 CH2 O 2b2 1b1 5a1 1b2 C2 H4 1b2u 1b2g 3ag 1b3u 2b1u NH3 3a1 1e C2 H2 1g 3 u 3 u
Table 2 Vertical ionization potentials (in eV) for the set of molecules, calculated using AUG-CC-PCVTZ basis set.
Ne 2p 2s HF 1 3 N2 3 g 1u 2 u CO 5 1 4 F2 1g 1u 3 g HCN 1 3 2
IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSDT
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
21.38 48.42
21.36 48.47
21.41 48.19
21.60 48.65
21.57 48.69
21.55 48.22
21.56 48.42
16.08 20.01
16.03 19.95
16.10 20.03
16.26 20.16
16.21 20.10
16.22 20.12
16.05 20.00
15.70 17.28 18.95
15.48 16.97 18.79
15.54 16.99 18.74
15.84 17.41 19.07
15.62 17.10 18.91
15.63 17.09 18.82
15.60 16.98 18.78
14.22 17.11 19.84
14.03 16.91 19.76
13.95 17.03 19.60
14.33 17.26 20.00
14.14 17.05 19.92
14.04 17.13 19.70
14.01 16.91 19.72
15.66 19.02 21.22
15.51 18.91 20.94
15.69 18.89 21.11
15.88 19.22 21.38
15.73 19.11 21.10
15.85 19.02 21.23
15.83 18.8 21.1
13.95 14.01
13.69 13.85 20.52
13.76 13.89 20.44
14.06 14.14
13.80 13.98 20.62
13.86 13.99 20.54
13.61 14.01 20.1
D. Bokhan et al. / Chemical Physics Letters 610–611 (2014) 173–178
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Table 2 (Continued)
H2 O 1b1 3a1 1b2 CH2 O 2b2 1b1 5a1 1b2 C2 H4 1b2u 1b2g 3ag 1b3u 2b1u NH3 3a1 1e C2 H2 1g 3 u 3 u
IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSDT
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS limit
Exp.
12.64 14.81 19.07
12.54 14.74 19.00
12.64 14.83 19.07
12.78 14.94 19.18
12.69 14.87 19.10
12.74 14.90 19.14
12.62 14.74 18.51
10.87 14.67 16.16 17.46
10.78 14.47 16.06 17.34
10.84 14.61 16.06 17.10
11.00 14.78 16.30 17.56
10.90 14.58 16.20 17.45
10.94 14.70 16.15 17.19
10.88 14.5 16.0 16.6
10.69 13.15 14.89 16.31 19.60
10.52 13.07 14.76 16.23 19.53
10.65 13.10 14.76 16.13 19.28
10.76 13.23 15.00 16.39 19.68
10.59 13.15 14.87 16.31 19.60
10.70 13.15 14.83 16.18 19.35
10.95 12.95 14.88 16.34 19.40
10.89 16.6
10.78 16.5
10.88 16.57
10.99 16.7
10.88 16.6
10.97 16.6
10.85 16.5
11.60 17.25 19.15
11.39 17.13 19.05
11.47 17.16 18.99
11.69 17.38 19.23
11.48 17.23 19.13
11.54 17.18 19.06
11.40 16.30 18.39
Table 3 ), mean absolute error Statistical measurements of the errors in ionization potentials with respect to experimental data: mean error (), maximum absolute error (max abs (abs ) and standard deviation (std ). IP-EOM-CCSD AUG-CC-PCVDZ AUG-CC-PCVTZ max abs AUG-CC-PCVDZ AUG-CC-PCVTZ
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
IP-EOM-CCSDT/CBS 0.15
0.02 0.17
−0.03 0.05
0.35 0.30
0.25 0.18
0.82 0.95
0.70 0.83
1.08 1.08
0.97 0.93
abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.20 0.21
0.23 0.17
0.37 0.31
0.28 0.21
std AUG-CC-PCVDZ AUG-CC-PCVTZ
0.29 0.27
0.29 0.28
0.27 0.26
0.29 0.27
0.67
0.20
0.25
IP-EOM-CCSDT/CBS values grows when more inner ionization is considered. This also suggest that orbital relaxation, which is more important in core states than in valence ones, is not being handled well enough as one would expect, lacking an exponential operator analogous to exp( T1 ) for R1 in the target state [27], and coupling between R3 and R2 equations.
The IP-EOM-CCSDT method is known to be very accurate and in its CBS limit results usually can be used as a reliable estimate of vertical ionization potentials. However, there are still some large deviations from experimental data even if a full-triples approach is used. Though there can still be questions about the role of R4 , which can be significant at times, see [6], equally likely errors derive
Table 4 ), mean absolute Statistical measurements of the errors in ionization potentials with respect to IP-EOM-CCSDT/CBS results: mean error (), maximum absolute error (max abs error (abs ), standard deviation (std ), and root mean square (RMS). IP-EOM-CCSD
IP-EOM(R3)-CCSD
IP-EOM-CCSD(F12)
IP-EOM(R3)-CCSD(F12)
AUG-CC-PCVDZ AUG-CC-PCVTZ
0.14 −0.02
0.26 0.11
−0.19 −0.15
−0.07 −0.02
max abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.37 0.27
0.33 0.34
0.39 0.37
0.33 0.26
abs AUG-CC-PCVDZ AUG-CC-PCVTZ
0.16 0.09
0.25 0.14
0.19 0.15
0.10 0.08
0.14 0.12
0.14 0.12
0.11 0.10
0.11 0.10
0.20 0.12
0.29 0.16
0.22 0.18
0.13 0.10
std AUG-CC-PCVDZ AUG-CC-PCVTZ RMS AUG-CC-PCVDZ AUG-CC-PCVTZ
178
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from the difficult experimental extraction of vertical IPs. This issue has been previously studied for C2 H4 by Yau et al. [28] where it is suggested that theory might offer better values. Another possible source of error may be related to vibronic effects, causing difficulties in the experimental definition of vertical ionization. Surprisingly, for both double and triple- bases IP-EOM-CCSD are closer to IP-EOM-CCSDT/CBS results than the corresponding values of IP-EOM(R3)-CCSD. Doubtless, this is due to cancellation of errors and with greater basis completeness via (F12), the R operbalance may be compromised making the truncation of the ator more apparent. Indeed, comparing root mean square errors (RMS) in Table 4 for the double- basis one can see that IPEOM(R3)-CCSD(F12) method is significantly more accurate then all other methods. For triple- bases, where F12 correction is less important, IP-EOM(R3)-CCSD(F12) and IP-EOM-CCSD have similar RMS. One would hope that even more accurate results may be obtained if all diagrams contributing to the R3 operator are included in the corresponding EOM equation, but the computational cost of such a method would be proportional to N7 . 4. Conclusions The IP-EOM(R3)-CCSD and IP-EOM(R3)-CCSD(F12) methods are implemented and tested on a set of different systems and ionized states. In most cases IP-EOM(R3)-CCSD(F12) results even at double level are close to those of IP-EOM-CCSD/CBS. Taking into account the fact that the computational cost of IP-EOM(R3)-CCSD(F12) method is proportional to N6 , its results can be used as a reference for the interpretation of experimental data of photoelectron spectroscopy. For larger systems double- calculations are only feasible, and, since IP-EOM(R3)-CCSD(F12) appears to be the most accurate at this level among all considered methods, it can be recommended for practical calculations. Some large deviations from experimental results are likely related to non-vertical ionization of these small systems, encouraging the development and application of analytical gradients for IP-EOM to facilitate the determination of the experimentally observed adiabatic values when high accuracy is required. Presented study indicates that inclusion of R3 (even in its approximated form) and short-range correlation effects via
Slater geminals more important then the presence of T3 operator in neutral-state calculation. (The differences between using CCSDT-3 instead of CCSDT for the ground state T3 are negligible.) This conclusion will likely change for more difficult, multi-reference examples, but for the systems studied here, all differences fall within the statistical error bars. Acknowledgements We thank University of Florida High-Performance Computing Center (UF HPC) for providing computational facilities. This work has been supported partly by the US Air Force Office of Scientific Research. References [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]
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