Relativistic coupled cluster method based on Dirac ... - Science Direct

6 May 1994

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical

Physics Letters 222 ( 1994) 82-87

Relativistic coupled cluster method based on Dirac-Coulomb-Breit wavefunctions. Ground state energies of atoms with two to five electrons Ephraim Eliav (Ilyabaev) a, Uzi Kaldor a, Yasuyuki Ishikawa b aSchool of Chemistry, TelAviv University, 69978 TelAviv, Israel b Department of Chemistry, University of Puerto Rico, P. 0. Box 23346, San Juan, PR 00931-3346, USA Received 10 January 1994;in final form 3 March 1994

Abstract A relativistic Foclc-space coupled cluster theory based on the Dirac-Coulomb-Breit wavefunctions has been developed and implemented, employing analytic basis sets of Gaussian-type functions. Relativistic all-order calculations including single and double excitations were performed for the ground states of the He and Be atoms and for the Ne (Z= lo), Ar (Z= 18) and Sn (Z= 50) ions with 2-5 electrons. Comparison is made with the Dirac-Coulomb and non-relativistic formulation and with available experimental results. The non-additivity of relativistic and correlation effects is discussed.

1. Introduction

In recent years, accurate experimental studies on highly ionized ions have sparked renewed interest in few-electron systems. The experimental studies on relativistic and quantum electrodynamic effects have been performed on few-electron systems by employing advanced ion sources to produce highly ionized high-2 species [ 1,2 1. This in turn has prompted increasingly accurate theoretical calculations on fewelectron systems. Although the nature of the electromagnetic force that binds atoms is well known, construction of an effective theoretical method that accurately accounts for both relativistic and electron correlation effects is a challenging problem for many-electron systems. In the last few years, relativistic many-body perturbation theory (MBPT), which accounts for both relativistic and electron correlation effects, has been developed by several groups [ 3-51 using basis sets of

local [ 3 ] or global [4,5] functions. Relativistic MBPT provides a powerful and systematic means to calculate the properties of many-electron systems. While the first few terms of the perturbation series suffice for accurate calculations of the ionized species, the method is less suitable for high-accuracy calculations on neutral atoms where higher orders of relativistic MBPT are non-negligible. The expressions for the higher-order perturbation corrections are so complicated and time consuming to evaluate numerically that relativistic MBPT calculations are rarely carried out beyond third order [ 3-5 1. One alternative to performing the finite-order MBPT calculations is to use all-order methods. One of the most promising among these all-order methods is the coupled cluster (CC) formalism [ 6,7]. The CC method provides electron correlation with high accuracy, and is widely applied to non-relativistic atomic and molecular calculations [ 71. Relativistic coupled cluster (RCC) theory, which simultane-

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E.E. Iiyabaevet al. /Chemical PhysicsLetters222 (1994) 82-87

ously accounts for relativistic and electron correlation effects to all orders, has been developed in the last few years by using numerical finite difference methods [ 81 and discrete basis sets of local [ 91 and global functions [ 10,111. Recently, we have developed a relativistic version of the multireference valence-universal Fock-space CC theory which is based on the no-pair Dirac-Coulomb-Breit Hamiltonian

1111. The present Letter provides a brief description of the relativistic CC method [ 111 and reports the relativistic correlation energies of atoms and ions with 2-5 electrons. The next section describes briefly our relativistic CC method, which treats the instantaneous Coulomb and the low-frequency Breit interactions as an integral part of the two-electron interaction in both the SCF and RCC frameworks within the algebraic approximation. The results of the prototype RCC calculations on few-electron systems are presented and discussed in the third section.

2. Method The relativistic many-body Hamiltonian cannot be expressed in closed potential form. An approximate relativistic Hamiltonian most commonly used is the time-honored ‘no-pair’ Dirac-Coulomb (DC) Hamiltonian [ 12,13 1, H + =p+

(

F MO+

c vj y+ 3 e-j >

(1)

where (in atomic units) hi, is the one-electron Dirac Hamiltonian, h,(i)=Cai’pi+C’(Bi-l)+~~l,,,(i)

2

(2)

and V, is the instantaneous Coulomb interaction between electrons Vi/=rzT’.

(3)

Here P+=L+( l)L+(2)...L+(n), with L+(i) the projection operator onto the space spanned by the positive-energy eigenfunctions of the DF operator [ 121. In this form, the no-pair Hamiltonian is restricted to contributions from the positive-energy branch of the DF spectrum. The no-pair DC Hamiltonian is deficient, being

both non-covariant and not sufficiently accurate for precision calculation of fine structure separations and binding energies of inner-shell electrons. Adding the low-frequency Breit interaction, B =_ 1al.a2+(al.r12)((X2.r12)/r:2 12 2 r12

2

(4)

to the instantaneous Coulomb operator introduces the leading effects of the transverse photon exchange and remedies the lack of covariance of the no-pair DC Hamiltonian [ 14 1. Inclusion of the Breit interaction results in a Hamiltonian which contains all effects through order (Y’ [ 141, where cr is the line-structure constant. The nopair Dirac-Coulomb-Breit (DCB ) Hamiltonian [I41 H+ =Sf+

L

1 h,(i)+

1 (rul +Bij) 2+ i>j >

(5)

provides a satisfactory starting point for calculations on many-electron systems in the sense that it treats the electrons relativistically, it treats the most important part of the electron-electron interaction nonperturbatively and it puts the Coulomb and Breit interactions on the same footing in Dirac-CoulombBreit (DCB) SCF and RCC calculations. In order to study the electron correlation induced by the Breit interaction, the instantaneous Coulomb and frequency-independent Breit interactions are treated as an integral part of the two-electron interaction in the DCB SCF and RCC calculations [ 111. The Fock-space coupled cluster method has been described before [ 151, and only a brief outline is given here. The basic method follows Lindgren’s [ 16 ] choice of a normal-ordered wave operator, Q=l+s+t{P}+...

.

(6)

S is the excitation operator describing connected sin-

gle, double, ... excitations,

+ 4 1 {a~afa,u~}s~+... ijkl

,

(7)

where a! and Ujare creation and annihilation operators,sj, s&, ...) are excitation amplitudes, and the curly brackets denote normal order with respect to a refer-

E.E. Zlyabaevet al. /Chemical PhysicsLetters222 (1994) 82-87

84

ence (core) determinant. The summation is carried out over connected terms only. The equations determining the excitation amplitudes in a complete model space may be derived from the generalized Bloch equation [ 16 1,

is, &I = (Q~~-~v,Lnn,

(8)

v.p PVi2P)

(9)

where P and Q are the usual projection operators, and Ho and V result from the partitioning of the Hamiltonian, H=H,+V.

(10)

The Fock-space method is designed to handle openshell states. A closed-shell reference determinant is selected, and the excitation operator for states with n valence holes (electrons removed from the reference) and m valence particles (electrons added) is denoted by ?Wrn). The total S operator may be written as SE

c c n

&cp,m).

m

(11)

Haque and Mukherjee [ 17 ] have shown that this partitioning allows for partial decoupling of the Fockspace CC equations. The equations for S(‘J) involve only SC”‘) elements with k< i and I< j, and the large system of non-linear coupled equations separates, therefore, into smaller subsystems which are solved consecutively. The equations for S(‘*‘) are first iterated to convergence, then S(O*‘)and S(l*O)are solved using the known S(O*O), and so on. This separation is exact and reduces the computational effort signiticantly. The current application involved the closedshell (0,O) sector for the two- and four-electron systems, and the one-valence-particle sector (0, 1) for the three- and five-electron ions. The coupled cluster method with single and double excitations (CCSD) is employed, truncating the expansion of S at s=s,

+s, .

(12)

Detailed accounts of the matrix DCB SCF formalism and its applications to many-electron systems have been given in previous publications [ 18 ] and are not repeated here. The RCC method has also been described [ 111. The speed of light was taken to be 137.037 au throughout this work. The non-relativistic limit was simulated by setting the speed of light to

c= 1O5au. The SCF calculations were repeated without the Breit term. This is the conventional DiracCoulomb (DC) SCF scheme [ 18 1. Basis sets of welltempered Gaussian-type functions reported by Huzinaga and Klobukowski [ 191 were used in ah the calculations. The correlation of the K- and Gshell electrons required more p and d basis functions, with higher exponents than those used in the SCF functions. These were taken from the s orbital well-tempered series [ 19 1. Table 1 describes the bases employed. The finite nucleus model discussed in ref. [ 5 ] was used. The atomic masses used for the He, Be, Ne, Ar and Sn atoms are, respectively, 4.0026, 9.0122, 20.183, 39.948 and 118.69. The RCC calculations were performed using partial-wave expansion up to L -= 2 (i.e. s, p, and d basis functions) throughout this work.

3. Results and discussion SCF and RCC calculations were performed on the ground states of atomic He and Be and the ions of Ne (Z= lo), Ar (Z= 18) and Sn (Z=50) with two to five electrons. Table 2 displays the SCF (I&), second-order perturbation theory (&) and coupled cluster (I&r,) energies for the two-electron systems with Z=2, 10, 18, 50. The second-order correlation of He is much smaller in magnitude than the all-order CC result indicating that higher-order MBPT corrections are important for neutral systems. For the ionized species, on the other hand, the second-order correlation energies are close to the all-order values confirming that the MBPT series converges rapidly for these systems and the second-order corrections approximate the total energy to high accuracy [ 5 1. The correlation energy induced by the Breit interacTable 1 Basis functions. Members of the well-tempered s-series of ref. [ 19 ] used in the different I sectors

He Be Ne Ar Sn

Basis

S

P

d

(20s15plld) (24s16p12d) (24s16p12d) (25s16p13d) (25s19p14d)

l-20 l-24 l-24 1-25 l-25

5-19 7-22 7-22 6-21 4-22

8-18 9-20 9-20 8-20 6-19

85

E.E. Ilyabaevet al. /Chemical PhysicsLetters222 (1994) 82-87 Table 2 Non-relativistic, Dirac-Coulomb

and Dirac-Coulomb-Breit

Atom

ground state energies of the two-electron systems (in hartree)

&C!F

E2

ECCSO

He (2~2)

NR DC DCB

-2.861679 -2.861813 -2.861750

-0.035662 -0.035660 -0.035693

-0.041081 -0.041081 -0.041112

Nes+ (Z= 10)

NR DC DCB

-93.861082 -93.982764 - 93.970656

-0.042698 -0.042654 -0.043901

-0.043961 -0.043929 -0.045169

Ar16+ (Z= 18)

NR DC DCB

-312.860572 -314.199585 -314.125608

- 0.043584 -0.043486 -0.047659

- 0.044299 -0.044221 - 0.048384

StP*+ (Z= 50)

NR DC DCB

-2468.800302 -2556.313923 -2554.613306

-0.044306 -0.045174 - 0.077044

-0.044568 -0.045484 -0.077338

tion is the difference between the electron correlation based on the DCB Hamiltonian and that based on the DC Hamiltonian, i.e. &,&DCB) EccsD( DC). While this correlation contribution is only 3.1 x 10b5 E,, for He, it increases dramatically for the high-2 Sn48+ ion to 3.2~ 10d2 E,,, in agreement with previous relativistic MBPT calculations

[51. Table 3 shows the SCF, second-order and coupled cluster energies of the ground state Be and Be-like ions. A behavior quite different from that of the twoelectron systems is observed for the correlation corrections in the highly ionized four-electron systems.

Table 3 Non-relativistic, DiraoCoulomb

and Dirac-Coulomb-Breit

Atom

Due to the relative phases of the large and small components of the Dirac four-spinors, the correlation energies of Sn46+ obtained by employing the DC and DCB Hamiltonians are much smaller than the nonrelativistic correlation energy. While the DCB correlation energy is significantly larger in magnitude than the DC correlation energy in the two-electron Sn4’+ ion, it is only slightly larger in the four-electron Sn46+ ion. The Breit interaction is a short-range interaction which, in the classical limit, behaves as re3. Therefore, the correlation energy induced by the Breit interaction in Sn46+ (0.041 E,,) is primarily due to the ls2 pair and is close to that of the two-electron

energies of the four-electron systems (in hartree)

-J&F

Be (Z=4)

NR DC DCB

- 14.573023 - 14.575892 - 14.575189

Ne6+ (Z= 10)

NR DC DCB

AP+

(Z= 18)

StP+ (Z=50)

E2

E CCSD

-0.072105 - 0.072097 -0.072277

-0.091893 -0.091883 -0.092059

-110.110978 - 110.255964 - 110.242097

-0.123493 -0.123277 -0.124719

-0.175863 -0.175136 -0.176549

NR DC DCB

- 377.543977 -379.198169 -379.111121

-0.176042 -0.173565 -0.178707

-0.273396 - 0.264629 -0.269678

NR DC DCB

- 3047.190429 -3159.832862 -3157.780188

-0.371400 -0.276310 -0.320144

-0.651927 -0.413677 -0.454224

86

E.E. Ilyabaev et al. /Chemical Physics Letters 222 (1994) 82-87

Table4 Koopmans’theorem (KT) and correlated(CCSD) 2s ionizationspotentialsof the three-electronions (in hartree) KT Ne (Z=lO)

NR DC DCB exp.a

Ar(Z=18)

NR DC

DCB exp.

CCSD

8.769853 8.782572 8.781698

8.775219 8.787946 8.787142 8.786720

33.585218 33.754616 33.748401

33.591279 33.760719 33.754778 33.7369

292.850327 305.604194 305.442366

292.856942 305.611393 305.452176

l

Sn (Z=50)

l

NR DC DCB

Refs. [21,22].

Sn4*+ ion (0.032 &), whereas the Coulomb correlation in the Be-like Sn46+ is an order of magnitude larger than in the two-electron ion. MBPT is known to give convergence problems for Be and Be-like ions due to quasi-degeneracy effects. The 1~~2s’singlet ground state of Be is strongly mixed with the 1s22p2 singlet configuration, and this makes it necessary to go to higher-order in MBPT to achieve accurate correlation energies. Thus, for all the fourelectron systems considered, the second-order energies evaluated by using the ls22s2 single configuration as a zeroth-order wavefunction without higher-

order contributions are far from the all-order results. Tables 2 and 3 show that the relativistic and correlation effects are strongly non-additive. Additivity would require similar E,--,, values for different SCF starting points applied to the same atomic system. While this is approximately true for the light atoms, changes of 55%-75% appear for the Sn ions. As noted above, these differences have opposite signs for the two Sn ions, leading to the conclusion that reliable estimates of relativistic and correlation effects require the simultaneous inclusion of both. The Fock-space coupled cluster method is particularly suitable for calculating ionization potentials and electron affinities. Here we calculate the energies of the ls22s and ls22s22p configurations of the Ne, Ar and Sn ions. Our total energies for the three-electron Ar (- 347.92877 Eh) and Sn (-2860.14282 E,,) are in good agreement with the numerical multireference Dirac-Fock values of Indicate and Desclaux [20] including the Breit term and excluding higher QED corrections, -347.92681 and -2860.13385 Eh, respectively. Table 4 gives the Koopman’s theorem and correlated ionization potentials of the three-electron systems. The former are simply minus the energies of the unoccupied 2s orbital of the two-electron ions, and the latter are the CCSD energies. The correlation contribution to the IP of Ne’+ is 5.4 m& in all three schemes. For Sn4’+, on the other hand, the non-relativistic correlation energy is 6.6 ml&, compared with 7.2 mEh for the DC and 9.8 m& for the DCB Ham-

Table5 Koopmans’ theorem(KT) and correlated(CCSD)Zp,,, ionizationpotentialsandZPfine structureof the five-electron ions (in

2P,,, IP KT

zP9,2-2P,,2 CCSD

KT

CCSD

NR DC

5.777264 5.782892

5.799046 5.804794

0.0 0.006663

0.0 0.006395

DCB exp. l

5.781703

5.803661 5.80386

0.006023

0.005799 0.005969

Ar (Z= 18)

NR DC DCB exp. ’

27.652203 27.770040 27.759158

27.647073 27.765008 27.754497 27.7731

0.0 0.110803 0.105413

0.0 0.106041 0.101049 0.103228

Sn (Z=50)

NR DC DCB

275.209990 286.566558 286.235213

275.082643 286.433495 286.107615

0.0 9.649423 9.483662

0.0 9.515328 9.355663

Ne (Z= 10)

* Refs. [21,22].

hartree)

E.E. Ilyabaevet al. /Chemical PhysicsLetters 222 (1994)82-87

iltonians. Correlation effects for the IP of the fiveelectron ions (Table 5) may have larger magnitude and different signs. Correlation contributes + 22 m&, to the IP of Nes+ by all three methods while the corresponding values for Ar”+ are - 5.2, - 5.0 and - 4.7 mE,, and for Sn4’+ -127, -133and-128m&The fine-structure splitting of the 2P state is zero in the non-relativistic limit. Its value is largely established at the DC-SCF stage but the Breit term and correlation also have significant effects (Table 5). The ionization potentials of the three- and five-electron Ne and Ar ions may be compared with experiment [ 2 1,22 1. Tables 4 and 5 show that the Ne values are within (2-4) x 10e4 E,, of the experimental IPs whereas the error in the Ar IPs is larger, about 0.0 18 &

Acknowledgement

This research was supported at TAU by the Israel Science Foundation administered by the Israeli Academy of Sciences and Humanities and by the USIsrael Binational Science Foundation. YI has been supported by the National Science Foundation through grant No. PHY-9008627. All grants are gratefully acknowledged.

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