Relativistic quasidegenerate perturbation theory with four-component general multiconfiguration reference functions

Relativistic quasidegenerate perturbation theory with four-component general multiconfiguration reference functions

Makoto Miyajima, Yoshihiro Watanabe, and Haruyuki Nakano^a兲

Department of Chemistry, Graduate School of Sciences, Kyushu University, Fukuoka 812-8581, Japan 共Received 18 October 2005; accepted 23 November 2005; published online 23 January 2006兲 Relativistic quasidegenerate perturbation theory 共QDPT兲using general multiconfiguration共GMC兲 reference functions is developed and implemented. It is the relativistic counterpart of the nonrelativistic QDPT with GMC reference and thus retains all the advantages of the nonrelativistic GMC reference QDPT, such as applicability to any configuration space and small computational cost compared to the complete configuration-space case. The method is applied to the potential-energy curves of the ground states of I₂ and Sb₂ molecules, the excitation energies of CH₃I, and the energies of the lowest terms of C, Si, and Ge atoms, and is shown to provide a balanced description of potential-energy curves and accurate transition energies for systems containing heavy elements and to provide much better results compared to the reference function 共i.e., active space configuration interaction兲level. ©2006 American Institute of Physics.

关DOI:10.1063/1.2161182兴

I. INTRODUCTION

The importance of simultaneous consideration of relativ- istic and electron correlation effects for describing the elec- tronic structures and chemical reactions of systems involving heavy atoms is now well recognized. Many methods for de- scribing electronic structures, including the electron correla- tion effect, have been transferred to the four-component rela- tivistic level: Møller-Plesset共MP兲perturbation, configuration interaction共CI兲, coupled-cluster共CC兲methods based on the Dirac-Hartree-Fock 共DHF兲 wave function, and the Dirac- Kohn-Sham method. Multireference共MR兲CI and CC meth- ods are also available through relativistic program packages such as DIRAC 共Ref. 1兲 and MOLFDIR.² However, as in the nonrelativistic case, the MRCI and MRCC methods require much computational cost. Lower cost multireference meth- ods are needed.

In the nonrelativistic case, multireference perturbation theory共MRPT兲based on multiconfiguration共MC兲reference functions has become a basic and practical tool for studying the electronic structures of molecules and the potential- energy surfaces of chemical reactions. Several versions of MRPT are now included in various program packages such asGAMESSandMOLCAS. MRPT takes account of both static and dynamic electron correlations and thus can obtain accu- rate relative energies, including reaction, activation, and ex- citation energies, within a chemical accuracy 共i.e., a few kcal/ mol兲.

We have developed an MRPT using MC functions that we call “multiconfigurational quasidegenerate PT 共MC-QDPT兲.”^3,4 It is a multiconfiguration basis multi- reference-state method based on van Vleck PT and includes multireference Møller-Plesset 共MRMP兲 PT,^5–7 a single- reference-state method based on Rayleigh-Schrödinger PT,

as a special case. In particular, a recently proposed version of MC-QDPT uses general multiconfiguration reference func- tions共GMC-QDPT or GMC-PT兲.^8,9GMC-QDPT imposes no restriction on the reference space, so it is much more com- pact than complete-active-space-共CAS兲based MRPT. In ad- dition, since it can avoid unphysical multiple excitations, it is numerically stable. In this article, we describe the extension of GMC-QDPT to a relativistic version with four-component general MC reference functions.

Other versions of relativistic MRPTs have been already presented. Vilkas et al. proposed a relativistic MRMP method based on multiconfiguration Dirac-Fock reference functions.¹⁰ Chaudhuri and Freed presented the relativistic effective valence shell 共H^v兲 method.¹¹These are relativistic generalizations of the nonrelativistic MRMP共Refs. 5–7兲and H^␯共Ref. 12兲methods. Vilkas and Ishikawa further developed a generalized relativistic MRMP method based on MRCI ref- erence functions.^13,14However, their applications have been limited to atomic systems—no molecular applications have been reported so far to the best of our knowledge.

In Sec. II, we briefly review GMC-QDPT and describe the relativistic GMC-QDPT and its implementation. We de- scribe the test applications of the scheme in Sec. III for the potential-energy curves of the ground states of I₂ and Sb₂ molecules, the excitation energies of methyl iodide CH₃I, and the energies of the lowest terms of C, Si, and Ge atoms.

In Sec. IV, we summarize the main points and make some concluding remarks.

II. METHOD

A. QDPT with general multiconfiguration reference functions

We briefly review GMC-QDPT in this subsection. The effective Hamiltonian matrix to the second order H_eff^共0–2兲 of van Vleck perturbation theory with unitary normalization¹⁵is given by

a兲Electronic mail: nakano@ccl.scc.kyushu-u.ac.jp

0021-9606/2006/124共4兲/044101/9/$23.00 124, 044101-1 © 2006 American Institute of Physics

共H_eff^共0–2兲兲MN=具⌽M共0兲兩H兩⌽N共0兲典+¹₂关具⌽M共0兲兩HR_NH兩⌽N共0兲典

+具⌽M共0兲兩HRMH兩⌽N共0兲典兴, 共1兲 with

R_M=_I苸Ref

兺

^{兩⌽I共0兲典共EM共0兲−EI共0兲兲−1具⌽I共0兲兩, 共2兲}

where⌽_M^共0兲共⌽_N^共0兲兲and⌽_I^共0兲are reference wave functions and a function in the complement space共Q兲of the reference space 共P兲, respectively, and E_M^共0兲 andE_I^共0兲 are zeroth-order energies of functions⌽_M^共0兲 and⌽_I^共0兲.

In GMC-QDPT, the reference wave functions␮共␯兲 are determined by MC-self-consistent field 共SCF兲 共or MC-CI兲 using general configuration space as an active 共variational兲 space, where the general configuration space 共GCS兲 is a space that is spanned by an arbitrary set of Slater determi- nants or configuration state functions共CSFs兲. Since the num- ber of reference functions is usually equal to the number of target states, the dimension of reference共P兲space is smaller 共in many cases, much smaller兲than that of GCS.

Taking the GCS-SCF or GCS-CI wave functions␮共␯兲as reference functions⌽_M^共0兲共⌽_N^共0兲兲, which define thePspace, Eq.

共1兲becomes

共H_eff^共0–2兲兲_␮␯=E_␮^GCS-CI␦␮␯

+1

2

再

^I苸GCS

兺

^{具␮E兩H兩I典具I兩H兩␯共0兲−EI共0兲␯典+共␮↔␯兲*}

冎

^,

共3兲 where I is now a determinant 共or a CSF兲 outside the GCS 共i.e., the active space兲. The notation 共␮↔␯兲^* means inter- change␮with␯and complex conjugation in the first term in the curly brackets. The complementary eigenfunctions of the GCS-CI Hamiltonian and the determinants共or CSFs兲gener- ated by exciting electrons out of the determinants共or CSFs兲 in GCS are orthogonal to the reference functions and define the Q space. The functions in the space complementary to thePspace in GCS, however, do not appear in Eq.共3兲since the interactions between the complementary functions and the reference functions are zero.

The third- and higher-order contributions are derived in the same manner,

共H_eff^共3兲兲_␮␯=1

2

再

^I,J苸GCS

兺

^{具␮兩H兩I典具I兩共V共E␮共0兲−EI共0兲−兲共EE␯共1兲␯共0兲兲兩J典具J兩H兩−EJ共0兲兲␯典}

+共␮↔␯兲^*

冎

^{,…, 共4兲}

though their computations are not very efficient compared to those for the second-order one, except for when the CI Hamiltonian matrix elements are readily available. The ef- fective Hamiltonian to the second order is therefore mostly used for applications to molecular systems.

B. Relativistic GMC-QDPT and implementation

The relativistic molecular theory without spinor optimi- zation can be derived along the same lines as in the nonrel- ativistic case if we begin with the no-virtual-pair Hamil- tonian,

H_DC⁺ =⌳⁺H_DC⌳⁺共⌳⁺=L⁺共1兲L⁺共2兲¯L⁺共N兲兲, 共5兲 for the Dirac-Coulomb Hamiltonian,

H_DC=

兺

_i ^hD共i兲+

兺

_i⬎j^{1/rij共hD共i兲=c␣·p+␤c2+Vnuc兲,}

共6兲 whereL⁺共i兲are the projection operators to electronic statesi,

␣ ^and␤ are Pauli matrices in the usual relativistic theory, andp andV_nucare momentum and nuclear attractive opera- tors, respectively. We can also add the Breit Hamiltonian,

H_B= −

兺

i⬎j

1

2rij

冋

^{␣i·␣j+ 共␣i·rijr兲共}ij^␣j·rij兲

2

册

^{, 共7兲}

toH_DCif necessary. Taking the second-quantized form of Eq.

共5兲, we get

H_DC⁺ =

兺

_pq ^hpqepq+1_4pqrs

兺

^{共pq储rs兲epq,rs}

=

兺

_pq ^hpqp+q+1_4pqrs

兺

^{共pq储rs兲p+r+sq, 共8兲}

which is the starting point of the relativistic GMC-QDPT.

The hpq are one-electron integrals for operator h_D共i兲, and 共pq^储rs兲 are antisymmetrized two-electron integrals 关共pq^储rs兲=共pq兩rs兲−共ps兩rq兲兴for operator 1 /rij.

Applying the same treatment used for obtaining Eqs.共3兲 and 共4兲 to the relativistic Hamiltonian, Eq. 共8兲, we get a formal expression for the relativistic GMC-QDPT,

共Heff兲_␮␯=E_␮^GCS-CI␦␮␯+1

2

再

^I苸GCS

兺

^{具␮兩HEDC+␯共0兲兩I典具I兩H−EI共0兲DC+ 兩␯典}

+_IJ

兺

苸GCS

具␮兩H_DC⁺ 兩I典具I兩共V_DC⁺ −E_␯^共1兲兲兩J典具J兩H_DC⁺ 兩␯典 共E_␮^共0兲−E_I^共0兲兲共E_␯^共0兲−E_J^共0兲兲

+共␮↔␯兲^*

冎

^{+…. 共9兲}

Reference functions␮are expanded using single Slater determinants兩A典as basis functions in our current implemen- tation,

兩␮典=_A

兺

苸GCS

C_A^␮兩A典. 共10兲

Other basis sets using alpha and beta strings as used in the MRCI method by Fleig et al.¹⁶ could also be used, but not implemented at present.

The second-order relativistic GMC-QDPT computation scheme is similar to that for nonrelativistic GMC-QDPT, which is described elsewhere.⁸We define the corresponding CAS 共CCAS兲 as a CAS constructed from the same active electrons and spinors, that is, the minimal CAS that includes

the GCS, and divide the summation over I in Eq. 共9兲 into summations over determinants outside CCAS and over the determinants outside the GCS but inside CCAS,

I苸

兺

GCS

=_I

兺

苸CCAS

+_I

兺

苸CCAS∧I苸GCS

. 共11兲

Using this division, we can write the second-order term in the curly brackets in Eq.共9兲as

共H_eff^共2兲兲_␮␯=_I

兺

苸CCAS

具␮兩H_DC⁺ 兩I典具I兩H_DC⁺ 兩␯典 E_␯^共0兲−E_I^共0兲

+_I苸CCAS∧

兺

_I苸GCS^具␮兩H_E^{DC+ 兩I典具I兩HDC+ 兩␯典}

␯共0兲−E_I^共0兲 . 共12兲 The first term in Eq. 共12兲 represents external excitations, while the second one represents internal excitations.

The external term can be expressed by 共H_external^共2兲 兲_␮␯=_AB

兺

_苸GCS^{CA␮*CB␯具A兩RHDC+} _E ^S

␯共0兲−H_DC^+共0兲H_DC⁺ R兩B典

=_AB

兺

苸GCS

C_A^␮*C_B^␯具A兩O^共2兲兩B典, 共13兲 whereRandSare projectors onto the CCAS and its comple- mentary space, respectively. OperatorO^共2兲 can be computed using the same diagrams used for the conventional QDPT with a CAS reference. Explicit formulas for this term are given in the Appendix. However, in GMC-QDPT关as well as in the original MC-QDPT 共Ref. 3兲兴 the rule for translating diagrams into mathematical expressions differs somewhat from that in conventional QDPT. The rule is described in detail elsewhere.⁹

In the diagrammatic computation of the effective Hamil- tonian matrix, the key idea is the particle-hole formalism, as is well accepted.¹⁷ In this formalism, particle-hole creation- annihilation operatorsb⁺ andb are used instead of electron creation-annihilation operatorsa⁺anda,

b⁺=a⁺, b=a for active and virtual spinors共“particles”兲, 共14兲 b⁺=a, b=a⁺for core spinors共“hole”兲,

where the state with all the core spinors occupied by elec- trons is taken as the vacuum state. The Hamiltonian in nor- mal form is expressed by

H_DC⁺ =E_core+

兺

_pq ^{fpqc 兵p+q其+1}_4pqrs

兺

^{共pq储rs兲兵p+r+sq其, 共15兲}

whereE_coreandf_pq^c are the core energy and core Fock matrix, respectively, and the curly brackets mean normal-ordered op- erators. This is the most commonly used expression.¹⁷How- ever, we can use another definition of particle-hole operators,

b⁺=a⁺, b=afor virtual spinors,

共16兲 b⁺=a, b=a⁺for core and active spinors,

where the state with all the core andactivespinors occupied by electrons is taken as the vacuum state. Holes in active spinors are treated as “quasiparticles,” while, in the represen-

tation of Eq. 共14兲, electrons in active spinors are treated as

“quasiparticles.” The Hamiltonian in normal form is given by

H_DC⁺ =Ecore+active+

兺

_pq ^{fpqca兵p+q其+1}_4pqrs

兺

^{共pq储rs兲兵p+r+sq其}

=Ecore+active+

兺

_pq^{共−fpqca*兲兵pq+其}

+1

4_pqrs

兺

^{共pq储rs兲*兵prs+q+其, 共17兲}

whereEcore+activeand f_pq^ca are the energy and Fock matrix for the occupied core and active spinors, respectively, and the asterisks denote complex conjugation.

A diagrammatic expansion to obtain an explicit formula for the effective Hamiltonian can be done based on either Eq.

共15兲or 共17兲. Since these equations are different expressions of the same Hamiltonian, they produced the same results.

However, the computational cost is different in general. In the representation of Eq.共14兲, the computation is done with coupling coefficients具I兩p⁺r⁺¯sq兩J典, wherep,q,r, andsare labels of active spinors. In the representation of Eq. 共16兲, coupling coefficients具I兩pr¯s⁺q⁺兩J典are used. Thus, if a cou- pling coefficient computation scheme such as the reduced intermediate space scheme of Zarrabianet al.¹⁸and Harrison and Zarrabian¹⁹ is used, the formalism using Eqs. 共14兲 and 共15兲is advantageous when the active spinors are filled with electrons less than the half number of active electrons, else the formalism using Eqs.共16兲and共17兲is advantageous. This is apparent from the fact that in these cases the reduced in- termediate spaces are smaller than those in the opposite cases. While the scheme in our current code differs from Zarrabian et al.and Harrison and Zarrabian, there are some similarities. The explicit formulas used for practical compu- tations are lengthy and therefore shown in the Appendix.

The internal term is computed with matrix operations for the Hamiltonian,

共H_internal^共2兲 兲_␮␯=

兺

I苸CCAS∧I苸GCS

⫻

冋

^A苸

兺

^{GCSCA␮*共HDC+ 兲AI•B苸GCS}

兺

^{共HE␯共0DC+兲−兲IBECI共0B␯兲}

册

^.

共18兲 Matrix elements 共H_DC⁺ 兲AI=具A兩H_DC⁺ 兩I典 共共H_DC⁺ 兲IB=具I兩H_DC⁺ 兩B典兲 may be readily available for the determinants in CCAS. The computational cost compared to the external term is negli- gible in most cases.

III. APPLICATIONS

We applied the present method to some molecular sys- tems to illustrate its performance. We calculated the potential-energy curves共PECs兲of the ground state of the I₂ and Sb₂molecules, the excitation energies of CH₃I, and the energies of the lowest terms of the C, Si, and Ge atoms.

Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians

were used for the molecules and for the atoms, respectively, and the spinors were determined with the Dirac-Hartree- Fock method usingDIRAC共Ref. 1兲 共for CH₃I兲,MOLFDIR共Ref.

2兲 共for C, Si, and Ge兲, and theREL4Dprogram²⁰of UTCHEM

共Ref. 21兲 共for I₂and Sb₂兲.

A. Potential-energy curves of I₂and Sb₂molecules We calculated the potential-energy curves of the ground states共X0_g⁺兲of I₂and Sb₂molecules, which are examples of single- and triple-bond dissociations, respectively. Two ac- tive spaces were taken for each molecule. The one for I₂was spanned by the determinants of which weights in the CAS共10,12兲 CI wave function were greater than 10⁻⁸ 共i.e., 兩C_I兩⬎10⁻⁴兲 and the other for I₂ was obtained using CAS共10,24兲instead of CAS共10,12兲, where CAS共n,m兲means the complete active space constructed from n electrons and m spinors. These GCSs are referred to as GCS共10,12兲 and GCS共10,24兲. The active spaces for Sb₂were similar to those for I₂, that is, the spaces spanned by the determinants se- lected from the CAS共6,12兲and CAS共6,24兲CI wave functions 关GCS共6,12兲 and GCS共6,24兲兴. The 12 spinors in CAS共n, 12兲 roughly correspond to 5p orbitals, and the additional 12 spinors in CAS共n, 24兲roughly correspond to diffuseporbit- als for flexibility of active spaces. The electrons in the lowest 56 spinors were not correlated 共56 frozen-core spinors兲 in the perturbation calculations for both molecules. We used Dyall’s VTZ basis set.²²

Table I summarizes the calculated spectroscopic con- stants for I₂. All the constants,re,␻e, andDe, of GMC-PT were in good agreement with the experimental values.²³ 共In this subsection, GMC-PT is used because the reference states are single.兲At the GCS-CI共i.e., reference function兲level, the differences from the experimental values for r_e,␻e, andD_e were 0.11 共0.08兲 Å, 51.5共46.5兲cm⁻¹, and 0.55 共0.73兲 eV, respectively, for GCS共10,24兲 关GCS共10,12兲兴. At the GMC-PT level, the differences were reduced to 0.03 共0.03兲 Å, 9.5共9.5兲cm⁻¹, and 0.17 共0.18兲 eV, respectively. The results of the Fock-space coupled-cluster method singles and doubles 共FSCCSD兲 method are also listed in Table I.

FSCCSD yielded very accurate values.²⁴The error trends of GMC-PT and FSCCSD共overestimation for re and underes- timation for ␻e and De兲 were similar, and thus the values produced by these methods were rather close.

Figures 1 and 2 show the ground-state PECs for the I₂ molecule obtained with second-order relativistic GMC-PT using GCS共10,12兲and GCS共10,24兲, respectively. The curves for DHF, second-order Møller-Plesset 共MP2兲 PT, and GCS-CI are also shown for comparison. The performance of the DHF and MP2 methods for radical breaking was similar to that of the corresponding nonrelativistic methods, that is, a good description in the equilibrium distance region and a poor one in the large bond distance region. In contrast, GCS-CI gave a qualitatively correct dissociation limit, and GMC-PT gave a quantitatively good description for the

TABLE I. Bond length, vibrational frequency, and dissociation energy for I₂ molecule.

Method r_e/ Å ␻e/ cm⁻¹ D_e/ eV

DHF 2.69 221 ¯

MP2 2.67 211 ¯

GCS-CI共10,12兲 2.75 168 0.83

GMC-PT共10,12兲 2.70 205 1.39

GCS-CI共10,24兲 2.78 163 1.01

GMC-PT共10,24兲 2.70 205 1.38

FSCCSD^a 2.69 214 1.47

Expt.^b 2.67 214.5 1.56

aReference 24.

bReference 23.

FIG. 1. DHF共䊐兲, MP2共쎻兲, GCS-CI共䊏兲, and GMC-PT共쎲兲potential-energy curves of the ground共X0_g⁺兲states of I₂. Active space GCS共10,12兲was used for GCS-CI and GMC-PT.

FIG. 2. DHF共䊐兲, MP2共쎻兲, GCS-CI共䊏兲, and GMC-PT共쎲兲potential-energy curves of the ground共X0_g⁺兲states of I₂. Active space GCS共10,24兲was used for GCS-CI and GMC-PT.

whole region, as the spectroscopic constants indicate. The effect of the active space was very small at the GMC-PT level.

Table II shows the spectroscopic constants of Sb₂, and Figs. 3 and 4 show the ground-state PECs. The behaviors of the methods were similar to those for the I₂molecule. How- ever, the differences for the spectroscopic constants were larger. The equilibrium nuclear distance at the GCS-CI level was 2.56 Å for both GCS共6,12兲 and GCS共6,24兲, 0.22 Å larger than the experimental value, 2.34 Å. This distance was not fully recovered at the GMC-PT level; it was still 0.18 共0.19兲 Å larger for GCS共6,24兲 关GCS共6,12兲兴. This was also the case at the MP2 level, which gave a similarreof 2.52 Å.

The difference between GMC-PT and the experimental val- ues in dissociation energy was also not small: 0.46共0.45兲eV larger than the experimental value, 3.09 eV, for GCS共6,24兲 关GCS共6,12兲兴. However, these features were not the only ones of the relativistic GMC-PT. A two-component method MRSDCI/RCI gave a 0.25 Å longer distance and a 1.23 eV larger dissociation energy.²⁵ Furthermore, scalar 共one-

component兲 GMC-PT with GCS共6,24兲 using the Martin/

Sundermann Stuttgart relativistic, large core valence triple- zeta effective core potential basis set^26,27 gave 2.56 Å , 221 cm⁻¹, and 2.08 eV. We do not pursue this issue further since the Sb₂ calculations were a part of the test ap- plications.

The selection of Slater determinants based on the CAS-CI coefficients means that different active spaces are taken depending on the molecular geometry, and this may cause PEC discontinuity. However, if a suitably small thresh- old is chosen, the PECs are mostly smooth, and the advan- tage of reducing the computational cost is much larger than the disadvantage. Based on our experience, a threshold be- tween 10⁻⁴ and 10⁻³ is appropriate for 兩C_I兩. In the present calculations, we used 10⁻⁴ to be on the safe side.

B. Excitation energies of methyl iodide CH₃I

In our calculations of the excitation energies of methyl iodide CH₃I, we used target states of 1E, 2E, 3E, 1A₂, and 2A₁ states, which come mainly fromn to␴^* single excita- tions. The basis set used was a valence triple-zeta plus double polarization basis set. The valence functions were contracted from the uncontracted relativistic Gaussian-type functions basis set by Koga et al.,²⁸ and the polarization functions were taken from Dunning’s correlation consistent polarized valence triple-zeta共cc-pVTZ兲basis set.²⁹Three ac- tive spaces 共GCS I-III兲 were tested: MRSD- 共GCS I兲 and MRS-type共GCS II and III兲, that is, spaces spanned by parent configurations plus singles and doubles 共GCS I兲 and parent configurations plus singles 共GCS II and III兲, where singles and doubles were made within the active-spinor space. One DHF configuration, four HOMO-LUMO configurations, and four second-HOMO-LUMO configurations were used as the parent configurations. The singles and doubles in space I were constructed from 12 electrons and 20 spinors, corre-

TABLE II. Bond length, vibrational frequency, and dissociation energy for Sb₂molecule.

Method r_e/ Å ␻e/ cm⁻¹ D_e/ eV

DHF 2.45 340 ¯

MP2 2.52 283 ¯

GCS-CI共6,12兲 2.56 219 2.09

GMC-PT共6,12兲 2.53 273 2.64

GCS-CI共6,24兲 2.56 225 2.31

GMC-PT共6,24兲 2.54 279 2.63

MRSDCI/RCI^a 2.59 246 1.86

Expt.^b 2.34 270 3.09

aReference 25.

bReference 23.

FIG. 3. DHF共䊐兲, MP2共쎻兲, GCS-CI共䊏兲, and GMC-PT共쎲兲potential-energy curves of the ground共X0_g⁺兲states of Sb₂. Active space GCS共6,12兲was used for GCS-CI and GMC-PT.

FIG. 4. DHF共䊐兲, MP2共쎻兲, GCS-CI共䊏兲, and GMC-PT共쎲兲potential-energy curves of the ground共X0_g⁺兲states of Sb₂. Active space GCS共6,24兲was used for GCS-CI and GMC-PT.

sponding to carbon 2s and 2p, hydrogen 1s, and iodine 5p orbitals. The singles in GCS II 共III兲 were constructed from 12 electrons and 24 共36兲 spinors, where more spinors had been included to take the spinor optimization effect into ac- count instead of the electron correlation effect by doubles.

The lowest 30 spinors were frozen in the perturbation calcu- lations.

The computed excitation energies are summarized in Table III. The spin-orbit 共SO兲 MCQDPT results³⁰ are also listed for comparison. SO-MCQDPT 共Ref. 31兲 is a two- component multireference multistate perturbation method proposed by Fedorov and Finley. At the GCS-CI共i.e., refer- ence function兲 level, the deviations in excitation energies among the three active spaces were somewhat large. In con- trast, at the GMC-QDPT level, they were very close to each other, regardless of the active spaces. The largest deviation was only 0.04 eV, indicating that, at the GCS-CI level, the description level differed depending on the active spaces, while, at the GMC-QDPT level, the balance of the descrip- tion was well recovered.

The experimental results of magnetic circular dichroism are available for 1E, 2A₁, and 3Estates,³² and they are also listed in Table III. We can see that GMC-QDPT reproduced the experimental values well. Taking the results for space I as

an example, we can see that the deviations from the experi- mental values were 0.04, 0.04, and 0.11 eV for the 1E, 2A₁, and 3Estates, respectively.

Table IV shows the approximate weight of the reference function occupied in the first-order perturbed wave function,

W_ref=

冋

^{1 +}

兺

^{␮ 兩D␮兩2具⌿␮共1兲兩⌿␮共1兲典}

册

⁻¹

⯝ 具⌿ref兩⌿ref典/具⌿ref+⌿^共1兲兩⌿ref+⌿^共1兲典, 共19兲 with

⌿ref=

兺

_␮ ^{D␮⌿␮ref, ⌿共1兲=}

兺

_␮ ^{D␮⌿␮共1兲, 共20兲}

where ⌿_␮^ref共=兩␮典兲 and⌿_␮^共1兲 are the reference and first-order wave functions for state␮, respectively, and兵D_␮其 is the ei- genvector of the effective Hamiltonian matrix 共H_eff^共0–2兲兲_␮␯ for state ␮. The weight, W_ref, is a measure of the quality of the reference wave functions, and the relative weight calculated for different states gives a measure of the balance of the calculation.

The nonreference weight,共1 −W_ref兲, can be further de- composed into pure internal, internal, and external weights, where the pure internal weight means the contribution from the active- to active-spinor excitations not included in the

TABLE III. Vertical excitation energies of methyl iodide CH₃I in eV.

State

GCS I^a GCS II^b GCS III^c

GCS-CI GMC-QDPT GCS-CI GMC-QDPT GCS-CI GMC-QDPT SO-MCQDPT^d Expt.^e

1E 4.87 4.09 4.73 4.06 4.45 4.07 4.16 4.13

2E 5.06 4.26 4.92 4.23 4.63 4.23 4.30 ¯

1A₂ 5.47 4.65 5.33 4.62 5.02 4.63 4.65 ¯

2A₁ 5.56 4.71 5.44 4.67 5.13 4.68 4.69 4.75

3E 5.93 5.06 5.80 5.03 5.50 5.05 5.03 5.17

aMRSD active space constructed from 12 electrons and 20 spinors.

bMRS active space constructed from 12 electrons and 24 spinors.

cMRS active space constructed from 12 electrons and 36 spinors.

dReference 30.

eReference 32.

TABLE IV. Reference and SD configurations weights in the first-order GMC-QDPT wave function in percent.

State

GCS I GCS II GCS III

Ref. SD^a Ref. SD^a Ref. SD^a

1A₁ 89.3 10.7

共0.0⫹0.1⫹10.6兲

88.8 11.2

共0.8⫹0.1⫹10.3兲

88.9 11.1

共2.2⫹0.3⫹8.6兲

1E 85.5 14.5

共0.0⫹0.1⫹14.4兲

85.4 14.6

共0.6⫹0.1⫹13.9兲

87.7 12.3

共2.1⫹0.3⫹9.9兲

2E 85.5 14.5

共0.0⫹0.1⫹14.4兲

85.3 14.7

共0.6⫹0.1⫹14.0兲

87.6 12.4

共2.2⫹0.4⫹9.8兲

1A₂ 85.1 14.9

共0.0⫹0.1⫹14.8兲

84.9 15.1

共0.6⫹0.1⫹14.4兲

87.6 12.4

共2.1⫹0.4⫹9.9兲

2A₁ 85.2 14.8

共0.0⫹0.1⫹14.7兲

84.9 15.1

共0.7⫹0.1⫹14.3兲

87.4 12.6

共2.3⫹0.3⫹10.0兲

3E 85.4 14.6

共0.0⫹0.2⫹14.4兲

85.2 14.8

共0.6⫹0.2⫹14.0兲

87.3 12.7

共2.2⫹0.5⫹10.0兲

aThe three numbers in parentheses in the SD configurations weight indicate pure internal, internal, and external excitation weights共see the text for more details兲.

GCS, the internal weight means the core- to active-spinor excitations, and the external weight means the contribution from the excitations involving virtual spinors. These num- bers are also listed共in parentheses兲in Table IV.

From Table IV we can see that the reference weights were fairly large, about 85%共84.9%-89.3%兲, and the differ- ences between the ground and excited states in the same active space were small 共⌬W_ref^max= 4.2%, 3.9%, and 1.6% for GCS I, II, and III, respectively兲. This means that the qualities of the wave functions were similar, i.e., well balanced, be- tween the ground state and excited states, which supports our excitation energy results. The slightly larger weights of the ground state were due to the use of spinors optimized for the ground state. One more feature we can see from the table is that the pure internal contribution was very small共less than 1%兲except for GCS III, which includes relatively many ac- tive spinors. This validates our choice of active spaces.

C. Energy of the lowest terms of carbon, silicon, and germanium atoms

As a final example, we calculated the lowest terms of group IV atoms, C, Si, and Ge. We included the Breit inter- action in the Hamiltonian since the magnetic terms are im- portant for obtaining accurate spin-orbit splitting. The basis sets used were the uncontracted relativistic Gaussian-type function basis sets by Koga et al.,²⁸ augmented byd and f polarization functions taken from Dunninget al. augmented cc-pVTZ basis set.^33–35The spinors were optimized using the open-shell DHF method with two electrons in six spinors.

The active spaces were constructed from four electrons and 16 spinors共double valence spinor space兲in all atoms, and, as in the calculations of I₂and Sb₂, only determinants satisfying 兩CI兩⬎10⁻⁴for at least one of the³P₀,³P₁,³P₂,¹D₂, and¹S₀ states were included in the active space 关GCS共4,16兲兴. The

K-shell spinors of C, theK- andL-shell spinors of Si, and the K-,L-, andM-shell spinors of Ge were frozen in the pertur- bation calculation.

The results are listed in Table V. The GMC-QDPT re- sults were in very good agreement with the experimental values.³⁶ The average and maximum errors were only 3.0%

and 4.7%, respectively. The GCS-CI results were also close to the experimental values, except for the ¹D₂ state, for which the error was 25.6%-36.3%. GMC-QDPT provided better results than the GCS-CI in almost all the cases. SO- MCQDPT also gave very good results, especially for the¹D₂ and¹S₀ states.³¹However, for the spin-orbit splitting for the

3P states, GMC-QDPT yielded better results. The maximum error of SO-MCQDPT for these states was 20.3% whereas that of GMC-QDPT was 4.7%.

We also performed wave-function analysis using the ref- erence weights for these atoms. In all the atoms, the approxi- mate reference weights exceeded 90%共96.2%-96.8% for C, 93.3%-94.3% for Si, and 91.3%-92.3% for Ge兲, and the dif- ferences between the states were very small, which supports the accuracy of our results.

IV. CONCLUSION

We have described relativistic GMC-QDPT, i.e., an ex- tension of nonrelativistic GMC-QDPT to a relativistic ver- sion with four-component general MC reference functions. It retains the advantages of the nonrelativistic GMC-QDPT:

flexible selection of configuration spaces, avoidance of un- physical multiple excitations, efficient computation using both diagrammatic and CI-matrix based methods, etc.

We applied our scheme to the calculations of the potential-energy curves of I₂and Sb₂ molecules, the excita- tion energies of CH₃I, and the energies of the lowest terms of

TABLE V. Energies of the lowest terms of C, Si, and Ge atoms in cm⁻¹. The values in the parentheses are the errors in percent from experimental values.

Term GCS-CI GMC-QDPT SO-MCQDPT^a Expt.^b

C

3P₀ 0 0 0 0

3P₁ 15.48共−5.6兲 15.69共−4.3兲 13.27共−19.1兲 16.40

3P₂ 40.49共−6.7兲 41.56共−4.2兲 39.57共−8.8兲 43.40

1D₂ 12 798.92共25.6兲 10 323.98共1.3兲 10250.81共0.6兲 10 192.63

1S₀ 20 952.87共−3.2兲 21 041.33共−2.8兲 21106.28共−2.5兲 21 648.01 Si

3P₀ 0 0 0 0

3P₁ 71.78共−6.9兲 74.55共−3.3兲 62.33共−19.2兲 77.11

3P₂ 208.06共−6.8兲 215.63共−3.4兲 181.26共−18.8兲 223.16

1D₂ 8 586.97共36.3兲 6 378.96共1.3兲 6 284.01共−0.2兲 6 298.85

1S₀ 14 882.64共−3.3兲 14 904.32共−3.2兲 14 752.15共−4.2兲 15 394.36 Ge

3P₀ 0 0 0 0

3P₁ 455.95共−19.2兲 532.61共−4.4兲 443.92共−20.3兲 557.13

3P₂ 1 170.59共−17.0兲 1343.99共−4.7兲 1 152.07共−18.3兲 1 409.96

1D₂ 9 172.17共28.7兲 7197.43共1.0兲 7 118.54共−0.1兲 7 125.30

1S₀ 16 528.28共1.0兲 16001.39共−2.2兲 16 286.67共−0.5兲 16 367.33

aReference 31.

bReference 36.

C, Si, and Ge atoms. Except for the Sb₂ case, the present method gave results close to the experimental values and comparable to the results of other highly correlated relativ- istic methods such as the Fock-space coupled-cluster method and the spin-orbit MC-QDPT. Wave-function analysis sup- ported our calculations. The reference weights were all large, and their deviations were small, which indicates the high quality and good balance of the calculations.

The multipartitioning Hamiltonian approach,^37,38 which allows different partitioning for different states and thus is more flexible than the present method, and the intruder state avoidance approach³⁹ have also been implemented and are now available. Although these approaches, using a different Hamiltonian partitioning, are not in the framework of GMC- QDPT, they can be easily included with small changes as options at the program level.

ACKNOWLEDGMENTS

The present research was supported in part by a Grant- in-Aid for Scientific Research 共Division B兲 from the Japan Society for the Promotion of Science and in part by CREST from Japan Science and Technology Agency.

APPENDIX: EXPLICIT FORMULAS FOR EXTERNAL TERM OF SECOND-ORDER RELATIVISTIC

GMC-QDPT

In the text, explicit formulas are not given for the exter- nal term, Eq.共13兲, of the effective Hamiltonian matrix. Here we present formulas that can be used for practical computa- tion of second-order relativistic GMC-QDPT for the reader’s convenience. The formulas are similar to those for nonrela- tivistic MC-QDPT,³ but somewhat different, particularly if we use the representation given by Eq.共16兲.

We have two formulas. One is for the common represen- tation, Eq.共14兲, and the other is for the less common repre- sentation, Eq.共16兲. We present the one for Eq.共16兲first.

The external term is expressed by zero- to three-body terms,

共H_external^共2兲 兲_␮␯=K_␮␯^0-body+K_␮␯^1-body+K_␮␯^2-body+K_␮␯^3-body, with

K_␮␯^0-body=_B

兺

苸GCS

CB␮^*CB␯

兺

m=1 2

Om,

K_␮␯^1-body= −

兺

BeGCS_pq

兺

_苸act^{具␮兩qp+兩B典CB␯}

兺

m=1 6

Sm,

K_␮␯^2-body=_B

兺

苸GCS_pqrs

兺

苸act

具␮兩qsr⁺p⁺兩B典CB␯

兺

m=1 7

Dm, and

K_␮␯^3-body= −_B苸GCS

兺

_pqrstu

兺

_苸act^{具␮兩qsut+r+p+兩B典CB␯}

兺

m=1 2

Tm. The zero- to three-body terms are composed of the fol- lowing terms

共a兲 Zero-body terms O₁= −

兺

i苸core,act

兺

e苸vir

共ie兲共ei兲

␧e−␧i+⌬EB␯,

O₂= −1 4_ij

兺

苸core,act_ef

兺

苸vir

共ie^储jf兲共ei^储f j兲

␧e−␧i+␧f−␧j+⌬EB␯. 共b兲 One-body terms

S₁= −

兺

e苸vir

共pe兲共eq兲

␧e−␧q+⌬EB␯,

S₂=_i

兺

苸core

共iq兲共pi兲

␧p−␧i+⌬E_B␯,

S₃= −_i苸core,act

兺

_e

兺

_{苸vir␧e−␧}^{共ie兲共ei}_i+␧p−^储_␧^pq兲_q+⌬EB␯^,

S₄= −_i

兺

苸core,act_e

兺

苸vir

共pq^储ie兲共ei兲

␧e−␧i+⌬EB␯,

S₅=1 2_i

兺

苸core,act_ef

兺

苸vir

共pe储if兲共ei储fq兲

␧e−␧i+␧f−␧q+⌬E_B␯,

S₆= −1

2_{ij苸core,act}

兺

_e

兺

_{苸vir␧e−}^共iq_␧i+^{储je兲共ei}_␧p−␧^储_j+^pj兲_⌬EB␯^.

共c兲 Two-body terms D₁= −1

2

兺

e苸vir

共pe兲共eq^储rs兲

␧e−␧q+␧r−␧s+⌬EB␯,

D₂=1 2_i

兺

苸core

共iq兲共pi^储rs兲

␧p−␧i+␧r−␧s+⌬EB␯,

D₃= −1 2_e

兺

苸vir

共pq储re兲共es兲

␧e−␧s+⌬E_B␯,

D₄=1

2_i苸core

兺

_␧^共pq_r−␧^储_i^{is兲共ri兲}_+⌬EB␯^,

D₅= −1

8

兺

ef苸vir

共pe^储rf兲共eq^储fs兲

␧e−␧q+␧f−␧s+⌬EB␯,

D₆= −_i

兺

苸core,act_e

兺

苸vir

共pq^储ie兲共ei^储rs兲

␧e−␧i+␧r−␧s+⌬EB␯,

D₇= −1 8_ij

兺

苸core,act

ⴱ 共iq^储js兲共pi^储rj兲

␧p−␧i+␧r−␧j+⌬E_B␯. 共d兲 Three-body terms

T₁= −1

4_e

兺

_{苸vir␧e−}^共pq_␧s+^{储re兲共es}_␧t−␧u^储₊^tu兲_⌬EB␯^,

T₂=1

4_i苸core

兺

_␧r−^共pq_␧i+^储_␧^is兲共ri_t−␧u^储₊^tu兲_⌬EB␯^.

Summation symbol兺^*inD₇means that the summation for theijpair is taken so that at least one ofi or jis a core-spinor label. The symbol共pq兲 denotes the Fock matrix,

共pq兲=f_pq^ca=hpq+

兺

i苸core,act共pq^储ii兲.

The⌬EB␯ represents the energy differences of zeroth- order configurationBand reference state␮,

⌬EB␯=E_B^共0兲−E_␯^共0兲=_p

兺

苸act

关具B兩p⁺p兩B典−具␯兩p⁺p兩␯典兴.

The computation is done using the coupling coefficient driven method. These coupling coefficients are sparse and can be prescreened based on the condition 具␮兩qs¯r⁺p⁺兩B典C_B^␯⬎␦^{, where} ␦^{= 10−9} is usually suffi- cient to keep the energy accuracy better than 10⁻⁵ hartree.

The formula for Eq. 共14兲can be obtained with the fol- lowing changes.

共1兲 Fock matrix

共pq兲=f^c_pq=hpq+_i

兺

苸core

共pq^储ii兲.

共2兲 Coupling coefficients 具␮兩qp⁺兩B典→−具␮兩p⁺q兩B典, 具␮兩qsr⁺p