Effect of three body transformed Hamiltonian (H∼3 ) through full connected triples on main and satellite ionization potentials computed via valence universal EIP-MRCC method

Chemical Physics Letters 474 (2009) 199–206

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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

e 3 ) through full connected triples Effect of three body transformed Hamiltonian ( H on main and satellite ionization potentials computed via valence universal EIP-MRCC method Kalipada Adhikari a, Sudip Chattopadhyay b, Ranendu Kumar Nath c, Barin Kumar De d, Dhiman Sinha c,* a

Department of Physics, M.B.B. College, Agartala, Tripura 799 004, India Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India c Department of Chemistry, Tripura University, Suryamaninagar, Tripura 799 130, India d Department of Physics, Tripura University, Suryamaninagar, Tripura 799 130, India b

a r t i c l e

i n f o

Article history: Received 7 November 2008 In final form 8 April 2009 Available online 11 April 2009

a b s t r a c t Valence universal multi-reference coupled cluster (VUMRCC) method via eigenvalue independent partie 3 on ionization potentials through full connected triple tioning has been applied to estimate the effect of H ð1;0Þ f3 Sð1;0Þ e 3 is constructed using CCSDT1-A model for the ground state calculation. H . H excitations S 3

3

involves n4vir n4occ operations that may lead to large time consumption. Our investigation on HF, HCl, N2 and CO molecules using cc-pVDZ and cc-pVTZ basis sets indicates that the above effect varies from f3 Sð1:0Þ is essential in high 0.001 eV to around 0.5 eV (11.5 kcal/mol), thus suggesting that inclusion of H 3

accuracy calculation. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Systematic incorporation of triple excitations, or simply triples, to assess electron correlations highly accurately in ground state and in ionized/excited states has now emerged as a field of much interest in quantum chemistry. Calculations employing MRCI  ller–Plesset perturbation method [1], complete fourth order Mø theory (MP4) [2], ADC(3)/ADC(4) method [3,4], SAC-CI (generalR) method [5–7], EOM-CC method [8,9] etc. are some of the frontier theories in this respect. In many cases, electronic difference energies such as ionization potentials (IP) are obtained within (or near to) 0.1 eV of experimental results. Apart from the development in this direction, theories designed explicitly to treat multi-reference (MR) systems have been under development since the early stage of CC theory. Valence universal (Fock space) multi-reference coupled cluster (VU-MRCC) theory [10–13] is a very effective MR method for computing energy differences, such as IPs, double ionization energies (DIP), electron affinity (EA) and excitation energies (EE). In VU-MRCC theories one universal wave operator is used for all sectors of Fock space. It works relative to a common ‘vacuum’ jUHF i, the open-shell systems are reached by changing the number of electrons from the reference state. Though originally developed as an effective Hamiltonian (Heff) based theory [10], it now has alternative forms. The effective

* Corresponding author. E-mail address: [email protected] (D. Sinha). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.04.012

Hamiltonian method which involves nonlinear CC equations is prone to severe numerical divergence problem borne out of the well known intruder states as one goes to determine the inner valence main IPs and the satellite IPs. The theory of incomplete model space [12] is one way to tackle the divergence problem to some extent. But this may not be adequate, considering the grave situation. Using eigenvalue independent partitioning (EIP) technique Mukherjee and co-workers [14,15] have developed a novel alternative form of the effective Hamiltonian-based VU-MRCC theory (termed as EIP-MRCC) which seems more like a CI-structure, despite its normally being non-hermitian and also having ‘hidden nonlinearity’ in CC variables. Using EIP technique, the nonlinear VU-MRCC equations and the equation for Heff for any (m-hole, n-particle) model space (complete model space or incomplete model space) are converted into the respective set of non-hermitian eigenvalue equations. Starting with a suitable guess eigenvector, the desired solution of each low-lying root of spectroscopic interest is arrived at iteratively through the use of stable root-search and root-homing procedures employed in eigenvalue problems (see Ref. [14]). Since the norm of the eigenvector of each root is preserved to unity in every iteration, the method is less prone to divergence problem as compared to even the VU-MRCC theory for incomplete model space. Thus, the inner valences IPs along with most of the satellite IPs which are impossible to obtain by solving the nonlinear CC equations are calculable by applying the EIP-MRCC method. It is noteworthy that the EIP-MRCC equations for IP (1-hole, 0-particle problem) and for EA (0-hole, 1-particle problem) are identical with the respective SRCC-LRT equations [16].

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In the initial work, Mukherjee and co-workers have done some pilot calculations at EIP-MRCCD level [14], where D represents ð0;0Þ doubles S2 ¼ T 2 in the ground state calculation (Fig. 1b), and douð1;0Þ means ionization with simultaneous single excitation bles S2 (Fig. 1c). At this juncture, it is worth mentioning that the IPs come out as direct difference energies, and they are size-intensive [13,15]. Then, on the basis of subsystem embedding condition (SEC) [10,12,13], DIPs related to Auger electron processes of HF molecule ð1;0Þ have been calculated using S2 only. Similarly, EEs of N2 have been ð1;0Þ ð0;1Þ ð0;1Þ calculated using S2 , and S2 , where the doubles S2 (Fig. 1d) represents electron attachment with simultaneous single excitation. Chaudhuri et al. [17] studied IPs of HF and H2O by EIP-MRCC method ð1;0Þ ð1;0Þ using singles S1 and doubles S2 and included the effect of conð1;0Þ nected triples S3 perturbatively correct up to third order. It is to ð1;0Þ ð1;0Þ be emphasized that in their work, the back coupling of S3 to S1 ð1;0Þ and S2 was ignored. Contrarily in the present work, the connected ð1;0Þ ð1;0Þ triples S3 and the doubles S2 are fully coupled through the Eq. (4a–c), thus taking into account higher order perturbative terms sysð1;0Þ tematically, if not fully. Since we have kept all holes ‘active’, no S1 appeared in the present calculations. In the ground state CC calculation too, the two works differ significantly. While in the former work, ð0;0Þ no connected triples S3 (=T3) was considered in the ground state CC calculation which was done at full CCSD level only, in our work, along with full CCSD, effect of connected triples has also been taken ð0;0Þ through the approximation S3 ¼ T 3 ¼ fVT 2 =Deg3 as per the CCSDT1-A model for ground state CC calculation, and iterations were repeated till convergence is reached. It is pertinent to mention the fact that the CCSDT1-A method is known to have an excellent intrinsic accuracy for a wide variety of applications in quantum chemistry. It is important to note that in our work, there is no approximation in ð1;0Þ S3 in case of ionization. Mitra et al. [18] studied DIPs related to Auð1;0Þ ger electrons of NH3, HCl and PH3 using multiple solutions of S2 obtained from eigenvectors of IP calculation at EIP-MRCCSD level. At ð0;0Þ (=T3) EIP-MRCCSDs level, which considers connected triples S3 by CCSDT1-A approach [19] for ground state, and full connected trið1;0Þ i.e., ionization with simultaneous double excitations ples S3 (Fig. 1e) for ionized states, Chattopadhyay et al. [20] calculated only the IPs of outer valence regions of N2 and CO using cc-pVXZ (X = D,T,Q) bases. Later on, Chattopadhyay et al. [21] have calculated and analyzed all valence main IPs along with outer valence shake-up and inner valence shake-up satellites of HF and HCl molecules using same kind of basis sets. Comparison of the results generated via EIPMRCCSDs with experiments are highly encouraging and the results of all main IPs, except the first two IPs above 20 eV of HCl, are within (or very near to) 0.1 eV of experimental values. At this point it is worth mentioning, the basic equations of EIPMRCCSDs. From VU-MRCC-Bloch equation [12,13], one gets for IP calculation,

n

o e ð1;0Þ þ P ð1;0Þ HS e ð1;0Þ P ð1;0Þ ¼ H e ð1;0Þ P ð1;0Þ HP eff n o n o ð1;0Þ e ð1;0Þ ð1;0Þ e ð1;0Þ ð1;0Þ e eff Pð1;0Þ P HP HS Q þQ ¼ Q ð1;0Þ Sð1;0Þ H n o n o e ð1;0Þ þ Rð1;0Þ HS e ð1;0Þ Pð1;0Þ ¼ Rð1;0Þ Sð1;0Þ H e eff Pð1;0Þ Rð1;0Þ HP

ð1a—cÞ

where the symbol ‘tilde’ means that the transformed Hamiltonian e is constructed using the CC amplitudes T e 1 ð Sð0;0Þ Þ and ( H) 1 ð0;0Þ e 2 ð S Þ only obtained at CCSDT1-A level of approximation for T 2 the ground state calculation. The projectors for the doublet model space P(1,0) and for the doublet virtual subspaces Q(1,0) and R(1,0) for N-electron system, respectively, are

Pð1;0Þ ¼

Mp X

j/a ðN  1Þih/a ðN  1Þj;

a¼1

Q ð1;0Þ ¼

ND X

j/pab ðN  1Þih/pab ðN  1Þj;

ð2a—cÞ

abp

Rð1;0Þ ¼

NT X

pq j/pq abc ðN  1Þih/abc ðN  1Þj

abcpq

where a, b, c . . . refer to occupied spin orbitals (holes) and p, q . . ., to unoccupied spin orbitals. MP, ND, NT are the dimensions of the P-, Qand R subspaces, respectively. ð1;0Þ

With Sð1;0Þ ¼ S2

ð1;0Þ

þ S3

ð3Þ ð1;0Þ

where all unfrozen holes are active so that no S1 calculation, the EIP-MRCCSDs equations become

appears in the

n o n o e ð1;0Þ e ð1;0Þ þ Pð1;0Þ HS e 2 ð1;0Þ P ð1;0Þ þ P ð1;0Þ HS e 3 ð1;0Þ Pð1;0Þ ¼ H Pð1;0Þ HP eff n o n o n o e ð1;0Þ þ Q ð1;0Þ HS e 2 ð1;0Þ P ð1;0Þ þ Q ð1;0Þ HS e 3 ð1;0Þ Pð1;0Þ ¼ Q ð1;0Þ Sð1;0Þ H e eff Pð1;0Þ Q ð1;0Þ HP 2 n o n o n o e ð1;0Þ þ Rð1;0Þ HS e 2 ð1;0Þ P ð1;0Þ þ Rð1;0Þ HS e 3 ð1;0Þ P ð1;0Þ ¼ Rð1;0Þ Sð1;0Þ H e eff Pð1;0Þ Rð1;0Þ HP 3 ð4a—cÞ

(Using the orthogonality between Q-space determinants and Rspace determinants in R.H.S. of 4b and 4c). Projection of Eq. (4a–c) from the left on h/a ðN  1Þjs; of all h/pab ðN  1Þjs and h/pq abc ðN  1Þjs respectively, and projection P ð1;0Þ three equations on j/a ðN  1Þis from the right, and writing 2 Pð1;0Þ ð1;0Þ ð1;0Þ and matrices in relation to the S2 and S3 amplitudes, 3 leads to

Y PP þ Y PQ

ð1;0Þ X

þ Y PR

ð1;0Þ X

2

Y QP þ Y QQ

ð1;0Þ X

3

þ Y QR

2

Y RP þ Y RQ

ð1;0Þ X 2

e ð1;0Þ ¼H eff

ð1;0Þ X

¼

3

þ Y RR

ð1;0Þ X 3

ð1;0Þ X

e ð1;0Þ H eff

ð5a—cÞ

2

¼

ð1;0Þ X

e ð1;0Þ H eff

3

Pð1;0Þ Pð1;0Þ Introducing 2 ¼ X QP X 1 ¼ X RP X 1 PP and PP , and following Refs. 3 [14,15], it leads to the EIP-MRCCSDs equations in matrix form,

2

3 2 3 Y PP Y PQ Y PR X PP 4 Y QP Y QQ Y QR 5 4 X QP 5 Y RP Y RQ Y RR ðNDT þMP ÞXðNDT þMP Þ X RP ðNDT þMP ÞXMP 2 3 X PP ð1;0Þ ¼ 4 X QP 5 ½EPP MP XMP X RP ðNDT þMP ÞXMP

ð6Þ

Fig. 1. Downward arrow lines represent ‘holes(h)’ and upward arrow lines, ‘particles(p)’, and all double arrow lines are ‘active’. (a) Normal ordered two body part (V) of ð1;0Þ Hamiltonian H, (b) two electron closed shell CC operators or doubles (D), (c) normal ordered two electron operators for shake-up ionization or doubles S2 , i.e., ionization ð0;1Þ for electron attachment, (e) normal ordered three electron operators for with simultaneous single excitation of electron, (d) normal ordered two electron operators S2 ð1;0Þ shake-up ionization or triples S3 , i.e., ionization with simultaneous double excitations of electrons. It is to be emphasized that the downward double arrow line beneath the S(1,0) vertex does not participate in the EIP-MRCCSDs matrix construction.

K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

201

ð1;0Þ

where NDT = ND + NT, and ½EPP  consists of the IPs as direct difference energies. In this Letter we are concerned with the effect of three body f3 Sð1;0Þ transformed Hamiltonian through full connected triples H 3

f3 Sð1;0Þ appears (Fig. 2) in IP calculation by EIP-MRCCSDs method. H 3 in the 3h2p-3h2p block of EIP-MRCCSDs matrix (Fig. 3) only. At this point, we emphasize that the dimension of EIP-MRCCSDs matrix increases enormously as compared to that of EIP-MRCCSD matrix because the inclusion of triples increases the N-electron functions manifold (for cc-pVTZ basis) so as to determine the eigenvalues by iterative Rettrup algorithm [22]. There are only f3 Sð1;0Þ all six Goldstone diagrams (Fig. 2) arising from the term H

Fig. 3. Various blocks of EIP-MRCCSDs matrix for IP calculation. Shaded blocks stand for EIP-MRCCSD matrix.

3

of which involve n4vir n5occ computer operations. On doing intermediate storage, the computer operations reduce to [ n3vir n4occ þ n3vir n5occ ] operations for diagrams (Fig. 2a, b, e and f) while they reduce to [ n4vir n3occ þ n4vir n4occ ] operations for the diaf2 g , the term may f3 is only fV T grams (Fig. 2c and d). Clearly, as H 3

be supposed to contribute very little as compared to the usual e ¼ V þ fV T f1 g þ 1 fV T f1 T f2 g . It is e Sð1;0Þ , where V f1 g þ fV T terms V 2 2 2 3 2 to be emphasized that the diagram in Fig. 4 requires the maximum number of computer operations n4vir n3occ amongst all the diagrams e 3 appears in 2h1p-2h1p, 3h2p-1h and e Sð1;0Þ . H representing V 3

3h2p-2h1p blocks also of the EIP-MRCCSDs matrix but none havð1;0Þ for ionized states. Some diagrams of ing connected triples S3 3h2p-2h1p block involve  n4vir n3occ operations while other such diagrams involve lower number of operations. All diagrams of these three blocks are always included in the calculation. A legitif3 S3 ð1:0Þ from mate question may now arise. What if we suppress H the computation assuming that their values are negligibly small? For illustration, moderately big molecules such as benzene, pyridine etc. when involve cc-pVQZ basis (or bigger ones), the speed f3 S3 ð1:0Þ is included. of computation may be stymied severely if H How much will the effect be on the accuracy of the results if it is excluded? This is main focus of the present work. For investigation, we have chosen 4 molecules two of which, HF and HCl, are single bonded while the other two, N2 and CO, are triple bonded, and moreover isoelectronic. Since we calculated all valence IPs and plenty of satellite IPs up to 45 eV, we have taken relatively smaller basis sets cc-pVDZ and cc-pVTZ, and did not take bigger basis as it would consume extraordinarily high time which our present computational facilities cannot tackle with. But more importantly, this seems to cause nothing serious. This is because the above two basis sets illustrate the principal feature of the work quite well, that is, the difference of the theoretical IP profiles with and without f3 Sð1;0Þ . the inclusion of H 3 2. Computational scheme ð0;0Þ

For ground state CC calculation, the effect of triples S3 (=T3) is estimated by using the CCSDT1-A model of Bartlett et al. [19]. In this model, T 3 ¼ fVT 2 =Deg3 (diag-1 in Fig. 4 of Ref. [19]) where V is the two body part of the exact Hamiltonian H in normal ordered form and T2 is the two body closed shell CC operator, which by

e, Fig. 4. Goldstone diagram relating to a two body transformed Hamiltonian V ð1;0Þ contracted with triples for ionization S3 .

nature is in normal ordered form, and the curly-bracket {} represents normal ordering. In the starting iteration of CCSDT1-A calculation, the converged full CCSD T2 amplitudes are used in the expression for T3. T3 is substituted in the T1 determining equation (Eq. (17) and diag. 6 in Fig. 2 of Ref. [19]) and in the T2 determining equation (Eq. (18) and diag. 6 in Fig. 3 of Ref. [19]), and then again the CCSD computer program is run till the convergence of the ground state correlation energy at the CCSDT1-A level is achieved. The resulting one-body and two body CC amplitudes are denoted e 2 , respectively, which only are stored and used to cone 1 and T by T e1 þ T e 2 ÞÞg. Thus, e ¼ fHðexpð T struct the transformed hamiltonian H e contains one-body (e e ) and the open diagrams of H f ), two-body ( V f three-body ( H3 ) parts only in the EIP-MRCCSDs calculation. (For the entire set of EIP-MRCCSDs diagrams for IP calculation, see Figs. 1 and 2 in our previous work [21]. For how the guess eigenvectors f3 S3 ð1;0Þ in are determined, see Refs. [14,20,21]). To see the role of H terms of magnitude, we did two kinds of computations named f3 S3 ð1;0Þ along with scheme-A and scheme-B. Scheme-A includes H the other usual diagrams for EIP-MRCCSDs matrix, whereas in scheme-B, this term is totally absent. The difference is discussed in the subsequent section.

3. Results and discussion In our calculations we have considered four chemically interesting and challenging molecules (such as HF, HCl, N2 and CO). We have used cc-pVDZ and cc-pVTZ basis sets (spherical Gaussians) and experimental equilibrium geometry in our computations. In our calculations, the basis sets were collected from: http:// www.emsl.pnl.gov:2080/forms/basisform.html. Experimental IPs is presented in the Tables with a view to realizing the reliability of our theoretical results only. Too accurate comparison is not

f3 ¼ fV T f2 g contracted with connected triples for ionization Sð1;0Þ . Fig. 2. Goldstone diagrams relating to the three body transformed Hamiltonian H 3 3

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K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

possible here because of the restraint of our starting basis sets. For that, approaching towards basis set saturation as much as possible is necessary. All outer-valence main IPs are presented in Table 1. Since independent particle model is valid here, same Koopmanns configuration appears while going from one basis to another. Naturally, there is same one-to-one correspondence between schemef3 S3 ð1;0Þ ) gives A and scheme-B also. Since scheme-A (as it includes H more accurate IP, from now on or unless otherwise explicitly mentioned, it will be assumed that a theoretical IP value relates to scheme-A only. For single bonded molecules HF and HCl, the conf3 S3 ð1;0Þ is small and the maximum value is 0.015 eV tribution of H (0.346 kcal/mol). For 2Pu state of triple bonded molecule N2, the differences in the case of cc-pVDZ and cc-pVTZ are 0.047 eV (1.083 kcal/mol) and 0.038 eV (0.876 kcal/mol) respectively which look relatively higher. However, this difference may also be interpreted as an evidence for convergence of results regarding to the order attained in treating electronic correlation within chemical accuracy [23]. For 2P state of CO, the difference (cc-pVTZ) 0.118 eV (2.721 kcal/mol) is significant in view of the correlation dynamics of outer valence electrons. We now go onto the inner valence region. We see that the sizes of the basis sets sometimes influence the IP-profiles of the same molecule in higher energy regions considerably. We first study the single bonded HF and HCl molecules, the IPs of which are presented in Table 2. The first 2R+ satellite of HF shows that f3 S3 ð1;0Þ contributes by an amount 0.039 eV (0.899 kcal/mol) for H both bases. For 2R+ main IP, the contribution is very small e.g., 0.006 eV (0.138 kcal/mol) (cc-pVTZ). In the next satellite, it is 0.064 eV (1.476 kcal/mol) (cc-pVDZ) and 0.091 eV (2.371 kcal/ mol) (cc-pVTZ). In 2P states, only one satellite IP appears consistently for both bases and the contributions are 0.238 eV (5.488 kcal/mol) and 0.149 eV (3.436 kcal/mol), respectively. Other satellites do not have the basis-to-basis correspondence. However, scheme-A to scheme-B correspondence is retained, which is based on the dominant configurations with expansion co-efficient 0.3 or more.

In the case of HCl molecule, the twinning states above 20 eV appear very close in values in both schemes for each basis. The maxf3 S3 ð1;0Þ is 0.011 eV (0.253 kcal/mol) (ccimum contribution of H pVTZ) in the two states. The IPs onwards are arranged on the basis of dominant configurations. If dominant configurations differ from basis-to-basis substantially, they are put in different rows in the tables. Thus, some IP values which appear in case of cc-pVDZ may not appear at all in case of cc-pVTZ, and vice versa. Similarly, an IP for a basis appearing in scheme-A may be absent in scheme-B, and vice versa. While in the first case it is due to basis-set effect, in f3 S3 ð1;0Þ . If for an IP, scheme-A to the second case it is due to H scheme-B correspondence is observed, only then it is possible to make a comment on the amount by which the IP has been shifted to what extent in scheme-B relative to Scheme-A. In other words, a f3 S3 ð1;0Þ can be made. For quite quantitative picture of the effect of H f3 S3 ð1;0Þ are significant. The highest a few IPs, the contributions of H contributions are 0.234 eV (5.488 kcal/mol) for IP 39.645 eV (ccpVDZ basis) and 0.439 eV (10.123 kcal/mol) for IP 40.209 eV (ccpVTZ basis). In 2P symmetry, the first satellite IP 28.246(0.005) eV for ccpVDZ basis and 28.218(0.003) eV for cc-pVTZ basis show 0.09 eV f3 S3 ð1;0Þ . It is and 0.084 eV respectively, as the contributions of H noteworthy, that this theoretical IP corresponds to the experimental IP 28.67 eV. There are examples of IPs of one scheme which have no correspondence in the other scheme. For example, 2R+ 40.139 (0.005) eV (cc-pVTZ, scheme-A), 2P 40.681(0.001) eV (ccpVDZ, scheme-B), 2P 43.151(0.001) eV (cc-pVTZ, scheme-B), and 2 P 45.102(0.007) eV (cc-pVDZ, scheme-B) may be quoted. The values mentioned in parenthesis are relative intensities along with IPs. In the inner valence regions of N2 and CO, the contributions of f3 S3 ð1;0Þ are more pronounced than in the outer valence region. H Table 3 enlists the main and satellite IPs of N2 with relative intensities up to 0.001. The first IP for both bases in the table correspond f3 S3 ð1;0Þ to the experimental IP 2 Rþ 29:4 eV. The contributions of H g

for this IP are 0.082 eV (1.891 kcal/mol) for cc-pVDZ basis and 0.083 eV (1.914 kcal/mol) for cc-pVTZ basis. The experimental IP

Table 1 Contribution of the diagrams for three body transformed Hamiltonian of 3h2p-3h2p block of EIP- MRCCSDs matrix (Fig. 3) to vertical ionization potentials (in eV) of outer valence region (relative intensities have been put in the parentheses). Molecule

HF

States

2

P

2

R+

HCl

2

P

2

R+

N2

Rþ g

2

2

Pu

Rþ u

2

CO

2

R+

2

P

2

R+

Basis

I II I II I II I II I II I II I II I II I II I II

Configurations

1p1 3r1 2p1 5r1 3r1 g 1p1 u 2r1 u 5r1 1p1 4r1

r(HF) = 1.7328 au; r(HCl) = 2.40832 au; r (N2) = 2.074 au; r(CO) = 2.13 au. Basis – I: cc-pVDZ; Basis – II: cc-pVTZ. a Ref. [24]. b Ref. [25]. c Ref. [26]. d Ref. [27].

EIP-MRCCSDs

Expt.

Scheme-A

Scheme-B

|Diff|

15.389(0.959) 15.900(0.949) 19.457(0.967) 19.840(0.957) 12.260(0.965) 12.565(0.956) 16.349(0.962) 16.551(0.953) 15.013(0.934) 15.383(0.929) 16.679(0.965) 16.946(0.960) 18.289(0.846) 18.614(0.848) 13.495(0.927) 13.826(0.925) 16.665(0.931) 17.062(0.949) 19.253(0.857) 19.503(0.860)

15.404(0.959) 15.915(0.949) 19.471(0.967) 19.851(0.957) 12.261(0.965) 12.562(0.956) 16.347(0.962) 16.547(0.953) 14.993(0.933) 15.363(0.929) 16.632(0.965) 16.908(0.960) 18.297(0.843) 18.620(0.845) 13.449(0.927) 13.782(0.922) 16.663(0.930) 16.944(0.925) 19.272(0.853) 19.522(0.857)

0.015 0.015 0.014 0.011 0.001 0.003 0.002 0.004 0.020 0.020 0.047 0.038 0.008 0.006 0.046 0.044 0.002 0.118 0.019 0.019

16.12a 19.79a 12.8b 16.66b 15.6c 17.0c 18.8c 14.00d 16.90d 19.60d

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K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

Table 2 Contribution of the diagrams for three body transformed Hamiltonian of 3h2p-3h2p block of EIP-MRCCSDs matrix (Fig. 3) to inner valence main and satellite vertical ionization potentials (in eV) of HF and HCl. States

Scheme-A

Scheme-B

|Diff|

Scheme-A

Scheme-B

|Diff|

HFa

2

34.774(0.057) 39.633(0.805) 41.748(0.069)

34.735(0.056) 39.645(0.800) 41.684(0.069)

0.039 0.012 0.064

35.473(0.068) 39.624(0.773) 42.341 (0.043) 44.975(0.037)

35.434(0.066) 39.630(0.771) 42.250(0.046) 44.947(0.037)

0.039 0.006 0.091 0.028

2

36.092(0.001) 38.005(0.010)

35.951 (0.001) 37.767(0.009)

0.141 0.238 38.584(0.010) 46.330(0.001)

38.435(0.007) 46.106(0.001)

0.149 0.224

24.210(0.239) 26.705(0.419) 30.840(0.008) 31.147(0.017) 32.512(0.001) 33.665(0.013) 36.219(0.002) 36.923(0.059) 38.350(0.190) 38.798(0.028) 40.139(0.005) 40.209(0.001) 41.032(0.019) 42.285(0.002)

24.202(0.233) 26.701(0.419) 30.826(0.046) 31.103(0.018) 32.481(0.001) 33.620(0.010) 36.167(0.002) 36.860(0.061) 38.292(0.179) 38.837(0.026)

0.008 0.004 0.014 0.044 0.031 0.045 0.052 0.063 0.058 0.039

39.770(0.001) 40.909(0.014) 42.345(0.002)

0.439 0.123 0.060

41.80*

43.314(0.022)

43.305(0.018)

0.009

45.80*

28.218(0.003) 34.484(0.021) 35.670(0.001) 36.892(0.001) 37.400(0.001) 38.587(0.026) 39.406(0.001) 41.506(0.002)

28.134(0.003) 34.463(0.019) 35.598(0.001) 36.844(0.001) 37.277(0.002) 38.617(0.025) 39.400(0.001) 41.285(0.001) 43.151(0.001) 43.075(0.002) 44.104(0.001)

0.084 0.021 0.072 0.048 0.123 0.030 0.006 0.221

28.67 33.69

R+

P

HClb

HClb

2

R+

2

P

Basis: cc-pVDZ

Expt.a,b

Mol

24.131(0.189) 26.979(0.539)

24.120(0.184) 26.987(0.553)

0.011 0.008

31.415(0.053)

31.377(0.055)

0.038

33.440(0.021)

33.430(0.021)

0.010

38.693(0.001) 39.645(0.001) 40.063(0.004)

38.675(0.001) 39.411(0.002) 40.014(0.003)

0.018 0.234 0.049

43.147(0.049) 43.191(0.025) 44.250(0.003) 44.371(0.011) 45.025(0.001)

43.034(0.063) 43.116(0.023) 44.264(0.007) 44.395(0.011) 44.965(0.001)

0.113 0.075 0.014 0.024 0.060

28.246(0.005) 35.475(0.001)

28.156(0.004) 35.441(0.001)

0.090 0.034

38.761(0.008)

38.710(0.009) 40.681(0.001) 41.621(0.005)

0.051

41.867(0.005)

42.582(0.001) 43.632(0.014) 45.619(0.006) *

Basis: cc-pVTZ

42.599(0.003) 43.576(0.012) 45.102(0.007) 45.590(0.005)

0.246

0.017 0.056

42.976(0.004) 44.700(0.002)

0.099 0.596

32–33 39.65

23.90 26.00 29.96 32.24 33.69 34.67 37.0*

37.00*

41.8*

45.80*

0.029

Symmetries of these states are unassigned in the experimental results. That is why, we put them in both symmetries. Ref. [24]. Ref. [25].

a

b

2

Rþg 38:0 eV is well reflected in both the bases with 38.524 eV

(0.468) (cc-pVDZ) and 38.731 eV (0.366) (cc-pVTZ). The contribuf3 S3 ð1;0Þ are 0.193 eV (4.451 kcal/mol) and 0.504 eV tions of H (11.622 kcal/mol), respectively. We note that the value corresponding to the cc-pVTZ basis is considerably high. Another reported experimental IP is 2 Rþ u 25:2 eV. Both the bases show similar theoretical values, an example being 25.519 eV (0.063) for cc-pVDZ basis and 25.913 eV (0.057) for cc-pVTZ basis. The conf3 S3 ð1;0Þ are significant, e.g. 0.075 eV (1.729 kcal/mol) tributions of H and 0.083 eV (1.914 kcal/mol) for cc-pVDZ and cc-pVTZ basis sets, respectively. Throughout the entire table, there are many moderf3 S3 ð1;0Þ contributes ately strong and weaker satellites in which H significantly. Table 3 also enlists the theoretical IPs of CO with relative intensities P0.01. There are a lot more IPs with smaller relative intensities which we do not report here. We must remember that the high lying theoretical IPs are largely basis set dependent until the saturation limit is approached. That is why, the positions and the relative intensities may not show basis-to-basis correspondence in the f3 S3 ð1;0Þ values of higher energy region. We therefore cite only the H

the first 4 IPs for the two bases in the table. For IP 24.070 eV (0.087) for cc-pVDZ basis and 24.375 eV (0.078) for cc-pVTZ basis, the conf3 S3 ð1;0Þ are 0.041 eV (0.945 kcal/mol) and 0.046 eV tributions of H (1.061 kcal/mol), respectively. For IP 30.366 eV (0.077) for cc-pVDZ basis, and 30.394 eV (0.067) for cc-pVTZ basis, the contributions 0.120 eV (2.767 kcal/mol) and 0.110 eV (2.537 kcal/mol), respectively, are significant. For IP 32.750 eV (0.099) for cc-pVDZ basis and 32.978 eV (0.153) for cc-pVTZ basis, the contributions are 0.131 eV (3.021 kcal/mol) and 0.149 eV (3.436 kcal/mol), respectively. For IP 34.719 eV (0.011) for cc-pVDZ basis and 33.617 eV (0.107) for cc-pVTZ basis, the contributions are 0.095 eV (2.191 kcal/mol) and 0.679 eV (15.588 kcal/mol), respectively. The experimental IP 2R+ 38.0 eV however should not be correlated with the theoretical IP 38.566 eV (cc-pVTZ) because of its low relative intensity of 0.017. For proper correlation between theoretical IP and experimental IP, use of a bigger basis is possibly necessary. It is especially noteworthy that for N2 and CO, the graphs of the experimental spectra of these two molecules [26,27] indicate the existence of many peaks in inner valence regions which agree well with our observations of too many theoretical IPs. Another mentionable point is that the calculations over the two molecules show

204

Table 3 Contribution of the diagrams for three body transformed Hamiltonian of 3h2p-3h2p block of EIP-MRCCSDs matrix (Fig. 3) to inner valence main and satellite vertical ionization potentials (in eV) of N2 and CO. States

N2a

2

Rþ g

Basis: cc-pVDZ Scheme-A

Scheme-B

|Diffl|

Scheme-A

Scheme-B

|Diff|

29.466(0.134) 30.487(0.006) 31.621(0.009)

29.384(0.115) 30.261(0.006) 31.408(0.008) 34.836(0.167) 35.436(0.007)

0.082 0.226 0.213

29.733(0.142) 30.756(0.005) 31.747(0.006) 35.673(0.208)

29.650(0.126) 30.533(0.005) 31.552(0.005) 35.228(0.273)

0.083 0.223 0.195 0.445

37.051(0.042)

36.742(0.014)

0.309

37.740(0.238) 38.331(0.517) 39.115(0.028) 39.290(0.037)

0.276 0.193 0.131 0.174

42.841(0.006) 44.646(0.003) 45.305(0.004)

0.344 0.221

38.731(0.366) 39.027(0.176) 40.245(0.004) 40.499(0.004) 40.579(0.005) 41.110(0.001) 41.214(0.001) 41.765(0.008) 42.228(0.003) 42.311(0.003) 42.326(0.026)

38.227(0.368) 38.877(0.124) 40.060(0.005) 40.261(0.004) 40.530(0.002) 41.057(0.002)

0.504 0.150 0.185 0.238 0.049 0.053

41.707(0.007) 42.206(0.002) 42.270(0.003)

0.058 0.022 0.041

43.205(0.002) 44.008(0.010) 44.553(0.002)

0.350 0.055

35.657(0.050) 38.016(0.114) 38.524(0.468) 39.246(0.049) 39.464(0.067) 42.791(0.009) 43.185(0.013) 44.867(0.006)

Pu

35.050(0.003) 38.265(0.006) 41.797(0.010)

34.915(0.003) 38.119(0.006) 40.570(0.013)

0.135 0.146 1.227

42.235(0.005)

44.929(0.001) 45.695(0.007)

44.725(0.002) 45.626(0.006)

29.4

0.221

44.358(0.001) 44.498(0.004) 44.622(0.016)

2

Expt.a,b

Basis: cc-pVTZ

45.160(0.007) 35.759(0.002) 39.392(0.005)

42.038(0.003) 42.076(0.003) 43.277(0.001) 0.204 0.069

44.094(0.015) 44.881(0.004) 44.995(0.001) 46.034(0.004)

44.641(0.001) 45.119(0.008) 35.656(0.002) 39.185(0.005) 40.598(0.001) 41.030(0.001) 41.056(0.001) 41.922(0.002) 42.778(0.001) 43.659(0.005) 43.740(0.005) 44.155(0.001) 44.727(0.005) 45.432(0.001)

0.041 0.103 0.207

0.116 0.499 0.354 0.726 0.437

38.0

K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

Mol

N2a

Rþ u

2

25.519(0.063) 34.043(0.062) 39.874(0.002) 40.703(0.024) 40.950(0.005) 43.561(0.001) 46.550(0.002)

COb

2

R+

24.070(0.087) 30.366(0.077) 32.750(0.099) 34.719(0.011) 36.524(0.066) 36.545(0.358) 36.887(0.272) 37.908(0.017)

39.693(0.224)

0.075 0.463

43.509(0.001) 43.837(0.001) 46.186(0.004)

0.052

24.029(0.091) 30.246(0.082) 32.619(0.117) 34. 624(0.011)

0.041 0.120 0.131 0.095

36.123(0.285)

0.422

36.873(0.056) 38.336(0.135) 38.461(0.215) 38.777(0.055) 39.051(0.038) 39.742(0.152) 39.716(0.021)

0.364

41.199(0.047)

25.830(0.057) 33.757(0.045)

0.083 0.465

41.052(0.024) 41.395(0.001) 42.717(0.001) 43.214(0.004)

0.615 0.406 0.039 0.400

45.946(0.002)

0.152

24.375(0.078) 30.394(0.067) 32.978(0.153) 33.617(0.107) 36.444(0.098)

24.329(0.081) 30.284(0.071) 32.829(0.140) 32.938(0.119) 36.113(0.181) 36.757(0.056) 36.889(0.183) 36.925(0.084) 38.057(0.105) 38.381(0.011)

0.046 0.110 0.149 0.679 0.331

38.143(0.072)

1.582

40.237(0.260) 40.987(0.029)

0.513

37.252(0.368) 37.707(0.038) 38.566(0.017) 0.039 0.060

39.569(0.193) 39.725(0.034) 40.284(0.018) 40.750(0.149)

0.049

0.363 0.782 0.509

42.411(0.017) 42.714(0.015)

39.721(0.252) 40.207(0.013) 40.196(0.018) 40.278(0.109) 41.234(0.013) 41.542(0.019)

25.913(0.057) 34.222(0.053) 38.747(0.001) 41.667(0.006) 41.801(0.001) 42.756(0.001) 43.614(0.001) 43.910(0.001) 46.098(0.002)

42.652(0.013) 42.712(0.012) 42.794(0.052) 0.035

41.353(0.016) 41.408(0.011) 41.480(0.060) 41.615(0.015) 43.437(0.013) COb

2

28.278(0.012) 29.388(0.010)

P

36.029(0.020)

35.896(0.029)

35.251(0.015)

35.256(0.019)

0.005

38.206(0.015)

37.519(0.013)

0.687

0.133

25.2

38.00 K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

38.422(0.212) 38.717(0.051) 38.828(0.116)

25.444(0.061) 33.580(0.044)

For CO, IPs with rel. int. P 0.01 only are presented. a Ref. [26]. b Ref. [27].

205

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K. Adhikari et al. / Chemical Physics Letters 474 (2009) 199–206

abundance of 3h-2p configurations having expansion coefficients of the order of 0.30.

4. Conclusion We have employed here EIP-MRCC method for an investigation e 3) over the effect of three body transformed Hamiltonian ( H ð1;0Þ through full connected triples S3 on main and satellite ionization potentials (IPs) of molecular systems such as HF, HCl, N2 and CO. e 3 is constructed based on high performing CCSDT1-A model for H triples in ground state energy calculation. The Goldstone diagrams f3 S3 ð1;0Þ appear only in the enormously large 3h2p-3h2p block for H of the EIP-MRCCSDs matrix for IP calculation. Some related diagrams involve n4vir n4occ computer operations which may require a high computation time for evaluation of IPs. In view of that, one may hesitate to include these diagrams when working with moderately big molecules such as benzene, pyridine etc., considering the size of one-electron basis and hence the size of N-electron basis. In order for substantiating the theoretical evidences, we took two starting bases cc-pVDZ and cc-pVTZ. Our calculations show that while for single bonded HF and HCl, the effect is sometimes more than 2 kcal/mol, for triple bonded and iso-electronic N2 and CO, the effect is considerably high and may even be more than f3 S3 ð1;0Þ is essential 10 kcal/mol. This suggests that inclusion of H in high accuracy calculations. Acknowledgement We are indebted to Professor D. Mukherjee for being a source of inspiration all along the work.

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