local relativistic exact decoupling - eth z...the consideration of relativistic effects is essential...

Local relativistic exact decouplingDaoling Peng and Markus Reiher Citation J Chem Phys 136 244108 (2012) doi 10106314729788 View online httpdxdoiorg10106314729788 View Table of Contents httpjcpaiporgresource1JCPSA6v136i24 Published by the American Institute of Physics Additional information on J Chem PhysJournal Homepage httpjcpaiporg Journal Information httpjcpaiporgaboutabout_the_journal Top downloads httpjcpaiporgfeaturesmost_downloaded Information for Authors httpjcpaiporgauthors

Downloaded 25 Mar 2013 to 12913211873 Redistribution subject to AIP license or copyright see httpjcpaiporgaboutrights_and_permissions

THE JOURNAL OF CHEMICAL PHYSICS 136 244108 (2012)

Local relativistic exact decouplingDaoling Penga) and Markus Reihera)

ETH Zurich Laboratorium fuumlr Physikalische Chemie Wolfgang-Pauli-Strasse 10CH-8093 Zurich Switzerland

(Received 15 April 2012 accepted 4 June 2012 published online 27 June 2012)

We present a systematic hierarchy of approximations for local exact decoupling of four-componentquantum chemical Hamiltonians based on the Dirac equation Our ansatz reaches beyond the triviallocal approximation that is based on a unitary transformation of only the atomic block-diagonalpart of the Hamiltonian Systematically off-diagonal Hamiltonian matrix blocks can be subjectedto a unitary transformation to yield relativistically corrected matrix elements The full hierarchy isinvestigated with respect to the accuracy reached for the electronic energy and for selected molecularproperties on a balanced test molecule set that comprises molecules with heavy elements in differentbonding situations Our atomic (local) assembly of the unitary exact-decoupling transformationmdashcalled local approximation to the unitary decoupling transformation (DLU)mdashprovides an excellentlocal approximation for any relativistic exact-decoupling approach Its order-N2 scaling can be fur-ther reduced to linear scaling by employing a neighboring-atomic-blocks approximation ThereforeDLU is an efficient relativistic method well suited for relativistic calculations on large molecules Ifa large molecule contains many light atoms (typically hydrogen atoms) the computational costs canbe further reduced by employing a well-defined nonrelativistic approximation for these light atomswithout significant loss of accuracy We also demonstrate that the standard and straightforwardtransformation of only the atomic block-diagonal entries in the Hamiltonianmdashdenoted diagonal localapproximation to the Hamiltonian (DLH) in this papermdashintroduces an error that is on the order ofthe error of second-order DouglasndashKrollndashHess (ie DKH2) when compared with exact-decouplingresults Hence the local DLH approximation would be pointless in an exact-decoupling frameworkbut can be efficiently employed in combination with the fast to evaluate DKH2 Hamiltonian in orderto speed up calculations for which ultimate accuracy is not the major concern copy 2012 AmericanInstitute of Physics [httpdxdoiorg10106314729788]

I INTRODUCTION

The consideration of relativistic effects is essential tothe proper understanding of the chemistry of heavy-element-containing molecules1 2 The Dirac equation provides therelativistic quantum mechanical description of a single elec-tron in the presence of external electromagnetic potentials Asthe Dirac operator comprises four-dimensional operators itseigenfunctions possess four components The Coulomb inter-action turned out to be sufficiently accurate for the descriptionof chemical phenomena and the corresponding many-electronHamiltonian is called the DiracndashCoulomb Hamiltonian

A relativistic method based on the DiracndashCoulombHamiltonian is called a four-component method It yieldsnegative-energy solutions which are pathologic and of no usein chemistry Also a large number of basis functions are re-quired to properly describe the negative-energy states Theseare two drawbacks of four-component methods that motivatedthe development of two-component methods which removethe negative-energy solutions and can in principle exactlyreproduce the results of four-component calculations2

Several relativistic two-component methods were devel-oped in the past decades One of the widely used approaches

a)Authors to whom correspondence should be addressed Electronicaddresses daolingpengphyschemethzch and markusreiherphyschemethzch

is the second-order DouglasndashKrollndashHess method (DKH2)3 4

It employs the free-particle FoldyndashWouthuysen (fpFW) (Ref5) transformation as well as sequential DouglasndashKroll6 trans-formations to decouple the four-component operator Higherorder7ndash10 and even arbitrary-order11ndash16 DKH methods havebeen developed The zeroth-order regular approximation(ZORA) (Refs 17ndash19) is another highly successful relativis-tic two-component method Within the ZORA framework itis particularly easy to implement the calculation of molecularproperties see Refs 20 and 21 for very recent examples andRef 22 for a review

In recent years it was shown that if the goal is to ex-actly decouple the four-component Hamiltonian matrix in-stead of the Hamiltonian operator the formulas of exactdecoupling can be directly obtained The BaryszndashSadlejndashSnijders (BSS) (Refs 23ndash26) method aims at exact decou-pling of the free-particle FW-transformed four-componentHamiltonian matrix by a matrix operator of the form derivedin Ref 27 The key matrix operator which is used to constructthe decoupling transformation matrix is obtained by solv-ing an iterative equation However invoking the free-particleFW transformation turns out to be not necessary for the con-struction of the exact-decoupling transformation in a one-stepprotocol The pioneering work of this one-step transforma-tion was provided by Dyall28ndash32 in the form of the so-callednormalized elimination of the small component (NESC)

0021-96062012136(24)24410811$3000 copy 2012 American Institute of Physics136 244108-1


244108-2 D Peng and M Reiher J Chem Phys 136 244108 (2012)

approach Later it was generalized to the so-called exact two-component (X2C) method by several groups33ndash44 Since theDKH method also employs matrix techniques to evaluate theDouglasndashKroll transformation it is also able to exactly decou-ple the four-component Hamiltonian matrix This has beenshown within the arbitrary-order approach11ndash14 16 45 For re-views of these developments see Refs 46ndash52

Even within a scalar-relativistic approximation wherethe spin-dependent operators are neglected the constructionof the relativistic Hamiltonian matrix dominates the step ofthe one-electron integral evaluation This is the case becausethe relativistic transformation is done within an uncontractedbasis set and involves many matrix manipulations such asmultiplication and diagonalization For a large molecule in-cluding only one or a few heavy atoms this procedure wouldwaste computational resources since the relativistic effects arehighly local and only significant for heavy atoms It is there-fore clear that a local relativistic method is desirable Evenfor a molecule containing only heavy atoms such a local rel-ativistic approach would be preferable as we shall see

The relativistic Hamiltonian operator is universal incoordinate space and a local approach is unfeasible at theoperator level However since most quantum chemical cal-culations apply a linear combination of atom-centered basisfunctions we may employ these basis functions to constructatomic projectors as it is done in charge and spin populationanalysis53 For those Hamiltonian matrix blocks for whichrelativistic corrections are important we shall derive a localrelativistic Hamiltonian matrix to evaluate it The locality isthen exploited in the basis-function space instead of coordi-nate space A crucial aspect to determine will be which blockof the Hamiltonian matrix requires a relativistic descriptionand how to evaluate it Undoubtedly the heavy-atom diag-onal blocks will require a relativistic description while therelativistic description of heavy-atom off-diagonal blocksdepends on their contribution to physical observables andon the feasibility of applying the relativistic description Forrelativistic transformation techniques such as DKH and X2Cit is impossible to apply the relativistic transformation to anoff-diagonal block alone without information about otherblocks It requires the information of a full square matrix toconstruct the relativistic transformation Such a difficulty dis-appears if one employs an operator-based relativistic methodsuch as ZORA The off-diagonal block of the Hamiltonianmatrix can be directly computed from relativistic integralsHowever the ZORA method provides a poor description ofatomic inner shell orbitals which carry the largest part of therelativistic effect In this sense the exact-decoupling methodsare the best candidate for applying a local relativistic schemesince they do not neglect important relativistic effects

All current exact-decoupling methods employ the trans-formation technique It is therefore not trivial to apply the rel-ativistic description to heavy-atom off-diagonal Hamiltonianmatrix elements However in this work we propose that if thelocal approximation is applied to the decoupling transforma-tion instead of the Hamiltonian matrix one could obtain re-sults which include the relativistic description to heavy-atomoff-diagonal blocks We also note that the main effort for theconstruction of the relativistic Hamiltonian matrix with exact-

decoupling methods is due to the evaluation of the decouplingtransformation matrices Therefore a local approximation tothe decoupling transformation would also lead to a significantreduction of the computational cost

II RELATIVISTICALLY LOCAL STRUCTURE OF THEHAMILTONIAN MATRIX

The following notation will be used throughout this pa-per Upper case labels A B C denote heavy atoms whichrequire a relativistic description while lower case labels ab c denote light atoms for which a (standard) nonrelativis-tic (NR) description can be assumed to be accurate enoughbased on numerical evidence compiled within computationalchemistry in the past decades To be more precise the formerlabels refer to nuclei with high nuclear charge numbers whilethe latter refer to those with low nuclear charge numbers (typ-ically with nuclear charges of less than two dozen protons) Ofcourse the distinction whether a nucleus is considered heavyor light bears some sort of arbitrariness and it will be madeon the basis of the accuracy required in a quantum chemicalcalculation In general there are then five different types ofHamiltonian matrix blocks HAA HAB HAa Haa and Hab thatneed to be considered

The evaluation of the relativistic Hamiltonian matrix withexact-decoupling approaches requires a set of square matri-ces They are the nonrelativistic kinetic energy matrix T

Tij = 1

2〈λi | p2|λj 〉 (1)

the external potential matrix V

Vij = 〈λi |V|λj 〉 (2)

and the overlap matrix S

Sij = 〈λi |λj 〉 (3)

as well as an additional relativistic matrix W

Wij = 1

4c2〈λi |σ middot pVσ middot p|λj 〉 (4)

(note that all expressions are given in Hartree atomic units inwhich the rest mass of the electron takes a numerical valueof one) In the above equations λi denotes the ith 2-spinoratom-centered basis function V the external potential c thespeed of light and σ the vector of Pauli spin matrices Thebasis functions λi may be grouped according to the atomicnucleus to which they are assigned which is key to the localapproaches discussed in the following The matrix represen-tation of the relativistic Hamiltonian is then evaluated as afunction of the above-mentioned matrices

H = H (T VW S) (5)

A straightforward idea for a trivial local approximation isto apply the evaluation of the relativistic Hamiltonian matrixonly to atomic (ie diagonal) blocks

HAA = H (TAA VAAWAA SAA) (6)

(instead of to the full matrix but where A may also representa group of atoms) and to ignore the relativistic correction to



all off-diagonal blocks

HAB = TAB + VAB (7)

Adding more atoms to a group A will improve the accuracybut the computational advances will be lost when a group Abecomes very large This approach was employed in Refs 54and 55 and explored in detail in Ref 56 for the low-orderDKH method We refer to it as the diagonal local approxi-mation to the Hamiltonian (DLH) matrix It is clear that theDLH approximation can also be applied to relativistic exact-decoupling approaches The DLH approximation works wellat large interatomic distances but the difference to a referenceenergy increases with shorter distance

However relativistic corrections to off-diagonal blocksare also important especially when interatomic distances areshort The necessity of applying the relativistic description tooff-diagonal blocks has already been discussed in Ref 57 witha so-called two-center approximation In Ref 57 the DKHtransformation was applied to pairs of atoms (A oplus B)

HAoplusB = H (TAoplusB VAoplusBWAoplusB SAoplusB) (8)

with

HAoplusB =(

HAA HAB

HBA HBB

) (9)

to obtain a relativistically corrected off-diagonal Hamiltonianmatrix HAB Unfortunately such a two-center approximationintroduces an inconsistent treatment to diagonal blocks Forinstance HAA can be obtained from either the pair (A oplus B) orfrom another pair (A oplus C)

So far we have only discussed the AA and AB blockswhich by definition require a relativistic description for bothatoms Since relativistic corrections for light atoms are verysmall and may be neglected Haa and Hab can be approximatedby the nonrelativistic Hamiltonian matrices

Haa = Taa + Vaa (10)

Hab = Tab + Vab (11)

A relativistic scheme for the heavyndashlight hybrid off-diagonalblocks HAa should however be considered explicitly

III LOCAL DECOMPOSITION OF THE X-OPERATOR

The general expression of exact-decoupling transforma-tions can be written as

U =(

Z+ 1radic1+XdaggerX

Z+ 1radic1+XdaggerX

Xdagger

minusZminus 1radic1+XXdagger X Zminus 1radic

1+XXdagger

)equiv

(ULL ULS

USL USS

)

(12)

Z+ and Zminus are two-component unitary operators and onlyZ+ is required for the evaluation of the electrons-onlyHamiltonian The X-operator generates the electronic small-component functions ϕ+

S from the large-component functionsϕ+

L

ϕ+S = Xϕ+

L (13)

Two-dimensional Hamiltonians are then obtained by apply-ing the unitary transformation of Eq (12) to blockdiagonalize(ldquobdrdquo) the four-dimensional Dirac-based Hamiltonian D

Hbd = UDU dagger (14)

It yields both the electrons-only Hamiltonian H (as the upperleft block in Hbd) and the one for negative-energy solutionsalthough the latter will be discarded

As discussed by Dyall in Ref 29 for the NESC methodthe BreitndashPauli approximation to the X matrix which is usedto construct the relativistic transformation matrix is

X = I (15)

where I denotes the identity matrix Such an approximation isnot variationally stable and thus cannot be used for variationalcalculations However one could employ this approximationfor light atoms only

Xaa = I (16)

since the corrections to light atoms are small and may notaffect variational calculations This idea was suggested inRef 42 for the X2C method but no results were presented

Although the lowest level approximation to X Eq (15)did not provide useful results the atomic approximation to theX matrix gave results with very small errors As discussed byDyall in Refs 30 and 31 the X matrix can be approximatedas the direct sum of atomic blocks

X = XAA oplus XBB oplus middot middot middot (17)

This approximation resembles the linear combination ofatomic four-spinors ansatz in four-component calculationsThe same approach was also employed in Ref 58 in the senseof an atomic approximation to the projection on electronicstates If the electronic states are further transformed to a two-component picture by a renormalization matrix this gives riseto a local X2C method As discussed in Refs 38 and 42 thelocal approximation to the X matrix Eq (17) works well forspectroscopic constants of diatomic molecules

However there are some drawbacks of the local X-matrixapproximation to the X2C method First it only reduces thecost for the evaluation of X matrix while the exact-decouplingtransformation matrices are still of molecular dimension asthe renormalization matrix is not atomic block diagonal evenif the X matrix is The computational demands for the evalua-tion of the relativistic approximation are thus still tremendousfor large molecules Second the X = I approximation did notprovide a satisfactory treatment of the Aa blocks since it maysuffer from variational collapse and Eqs (10) and (11) cannotbe fulfilled within this approximation

IV LOCAL APPROXIMATIONS TO THEEXACT-DECOUPLING TRANSFORMATION

In this paper we suggest a local approximation to theexact-decoupling transformation We may approximate theunitary transformation by taking only the ldquoatomicrdquo diagonalblocks (all off-diagonal blocks are then set to zero)

U = UAA oplus UBB oplus middot middot middot (18)



where the atomic unitary transformations UAA are obtainedfrom the diagonalization of the corresponding DAA blocks ofmatrix operator D We denote this approximation to U as thediagonal local approximation to the unitary decoupling trans-formation (DLU)

If we now substitute the approximate unitary transforma-tion of Eq (18) into Eq (14) the diagonal blocks

HbdAA = UAADAAU

daggerAA (19)

turn out to be the same as in the DLH approach while theoff-diagonal blocks then read

HbdAB = UAADABU

daggerBB (20)

which is to be compared with the expression

HbdAB =

sumIJ

UAIDIJ UdaggerJB (21)

where U has not been approximated (here I and J run overall atomic blocks) Hence the DLU approach also introducesa relativistic description to off-diagonal ldquointeratomicrdquo blockswhen compared with the DLH approximation

The cost for the assembly of the unitary transformationU within the DLU approach is then of order N where Nmeasures the system size namely the number of atoms Itis linear scaling since the atomic approximation is directlyapplied to the unitary transformation However the nextstep of applying the unitary decoupling transformation toobtain the relativistic Hamiltonian matrices is no longerlinear scaling If no local approximation was applied forthe unitary decoupling transformation according to Eq(21) where the summation indices I J run over all atomicblocks the calculation of HAB matrix will require 2N2 matrixmultiplications Since the number of matrices to be calculatedis of order N2 the total cost of the relativistic transformationwithout local approximation is therefore of order N4 If theDLU approximation is applied no summation will be neededand only two matrix multiplications will be required for eachheavyndashheavy block HAB The total cost is then of order N2

If the distance of two atoms A and B is sufficiently largethe relativistic description of HAB can be neglected Thus wemay define neighboring atomic pairs according to their dis-tances Then only the Hamiltonian matrix Hbd

AB of neighbor-ing pairs requires the transformation

HbdAB = UAADABU

daggerBB forallAB being neighbors (22)

whereas all other pairs are simply taken in their nonrelativis-tic form Since the number of neighboring pairs is usually alinear function of system size the total cost is then reduced toorder N

To investigate the relativistic correction to hybrid HAa

blocks of the electrons-only Hamiltonian H we need to con-sider the explicit structure of the unitary decoupling transfor-mation

H = (ULL ULS

) (V T

T (W minusT )

) (ULLdagger

ULSdagger

) (23)

where only the electronic part of the exact-decoupling trans-formation represented by ULL and ULS [which are the upper

part of Eq (12)] is employed From Eq (23) it is easy to seethat the electrons-only Hamiltonian reads

H = ULST ULLdagger + ULLT ULSdagger minus ULST ULSdagger

+ULLV ULLdagger + ULSWULSdagger (24)

In the NR limit where the speed of light approaches in-finity we have

ULLNR = I ULSNR = I and WNR = 0 (25)

If we insert them into the relativistic electrons-only Hamilto-nian (24) we arrive at the nonrelativistic Hamiltonian matrix

H NR = T + V (26)

This suggests a solution for the relativistic treatment of thehybrid heavyndashlight blocks HAa because we may set all light-atom diagonal blocks in the unitary transformation to identitymatrices

ULLaa = I and ULS

aa = I (27)

To reproduce the nonrelativistic Hamiltonian matrices oflight-atom-only blocks we must set the W matrix of lightatoms to zero

Waa = 0 (28)

The WAa matrices are also set to zero

WAa = 0 (29)

We denote this approximation for the treatment of hybridblocks as the DLU(NR) approach If we do not introduceany approximations to the W matrices the transformationwill yield a Hamiltonian that differs from the nonrelativisticHamiltonian matrix on the order of 1c2 which correspondsto the order achieved in the BreitndashPauli approximation Wetherefore denote it as the DLU(BP) approach

The final local approximation to the unitary matrices isthen

ULL = ULLAA oplus ULL

BB oplus middot middot middot ULLaa oplus ULL

bb oplus middot middot middot (30)

ULS = ULSAA oplus ULS

BB oplus middot middot middot ULSaa oplus ULS


where the atomic unitary transformation matrices of heavyatoms are evaluated by the relativistic exact-decoupling ap-proaches The explicit expressions of Hamiltonian matrixblocks HAA and HAB are

HAA = ULSAATAAU

LLdaggerAA + ULL

AATAAULSdaggerAA

minusULSAATAAU

LSdaggerAA + ULL

AAVAAULLdaggerAA

+ULSAAWAAU

LSdaggerAA (32)

HAB = ULSAATABU

LLdaggerBB + ULL

AATABULSdaggerBB

minusULSAATABU

LSdaggerBB + ULL

AAVABULLdaggerBB

+ULSAAWABU

LSdaggerBB (33)



If the DLU(NR) approach is employed the unitary transfor-mation matrices of light atoms are simply set to identity ma-trices and the explicit form of HAa Haa and Hab is

HAa = ULLAATAa + ULL

AAVAa (34)



In the DLU(BP) approach however they read


AAVAa + ULSAAWAa (37)

Haa = Taa + Vaa + Waa (38)

Hab = Tab + Vab + Wab (39)

The relativistic transformation of a property operatorP13 14 16 45 59 60 requires an additional matrix Q and its ma-trix elements are given by

Qij = 1

4c2〈λi |σ middot pPσ middot p|λj 〉 (40)

The relativistic corrected property matrix then reads

P = ULLPULLdagger + ULSQULSdagger (41)

V DLU EVALUATION WITHIN EXACT-DECOUPLINGAPPROACHES

A The X2C approach

The X2C method employs

UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger

minus 1radic1+XXdagger X

1radic1+XXdagger

) (42)

as the exact-decoupling unitary transformation The unitaryoperators for the electronic states are then

ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL

X2CXdagger = 1radic1 + XdaggerX

Xdagger (44)

For an actual implementation we need their matrix represen-tations in a finite non-orthogonal space of the atom-centeredbasis functions The X matrix is evaluated by diagonalizingthe following generalized eigenvalue equation(

V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+

S denote the large- and small-componentmolecular-orbital coefficients respectively We obtain the Xmatrix by the following equation

X = C+S

(C+

L

)minus1 (46)

The matrix representation of ULL and ULS is then given by

ULLX2C = S12(Sminus12SSminus12)minus12Sminus12 (47)

ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)

In our atomic approximation to the unitary transformation Uwe need to evaluate the above equation for each ldquoatomicrdquoblock of the original Hamiltonian in order to obtain the di-agonal blocks ULL

AA and ULSAA The decoupling transformations

are then assembled according to Eqs (30) and (31) and therelativistic Hamiltonian matrix follows from Eq (23)

B The BSS approach

In the BSS approach all matrices are transformed to anorthonormal basis-function space first It is convenient to cal-culate the transformation matrix K by diagonalizing the non-relativistic kinetic energy matrix

T K = SKt (50)

since the eigenvalues t can be used for later evaluation ofthe free-particle FW transformation The eigenvector matrixK has the following properties

KdaggerSK = I and KdaggerT K = t (51)

The fpFW transformation features four diagonal block matri-ces

U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)

with E0 = radic2tc2 + c4 and f =

radic2c2t It is then applied to

yield a transformed four-dimensional Hamiltonian matrix D0

D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)

Next the exact-decoupling BSS transformation

U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger

minus 1radic1+RRdagger R

1radic1+RRdagger

) (54)

which has the same structure as the exact-decoupling trans-formation UX2C is applied The R matrix is obtained by diag-onalizing the free-particle FW-transformed four-componentHamiltonian matrix D0 and employing a similar relation tothe one in Eq (46)

After the exact-decoupling BSS transformation has beencarried out the Hamiltonian matrix is back-transformed to theoriginal (non-orthogonal) basis

H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)



Consequently we obtain the matrix form of ULLBSS and ULS

BSSfor the BSS approach as

ULLBSS = (Kminus1)dagger(1 + RRdagger)minus

12 (ULLfpFW + RdaggerUSLfpFW)Kdagger

(56)

ULSBSS = (Kminus1)dagger(1 + RRdagger)minus

12 (ULSfpFW + RdaggerUSSfpFW)Kdagger

(57)

The expressions for the relativistic Hamiltonian matrix (andproperty matrices) are then as given above

C The DKH approach

The DKH approach features the same initial transforma-tion as the BSS approach to obtain the transformed Hamil-tonian matrix D0 Subsequent decoupling transformations areexpressed as

U (m) =1prod

k=m

Uk (58)

with the generalized parametrization of the Uk9

Uk =[mk]sumi=0

akiWik (59)

where Wk are anti-Hermitian matrix operators and m is re-lated to the order of the DKH expansion Since the expressionof Wk is too lengthy to be presented here we refer the readerto Ref 16 for details If m is large (strictly if it approaches in-finity) exact decoupling is achieved Usually a low value form is often sufficient for calculations of relative energies andvalence-shell properties (ie m = 1 for the original DKH2approach)

With the complete decoupling DKH transformation writ-ten as

UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)

the matrix forms of ULLDKH and ULS

DKH are

ULLDKH = (Kminus1)dagger(ULL(m)ULLfpFW + ULS(m)USLfpFW)Kdagger

(61)

ULSDKH = (Kminus1)dagger(ULL(m)ULSfpFW + ULS(m)USSfpFW)Kdagger

(62)

We should note that the finite-order DKH Hamiltonianmatrices are not the result of directly applying the decouplingtransformation To give consistent results the transformationin Eq (23) is only used to obtain the off-diagonal blocks whilethe diagonal blocks are replaced by the traditional finite-orderDKH Hamiltonian However for high-order calculations thedifferences of withwithout replacing the diagonal blocks arevery small

VI RESULTS AND DISCUSSION

As we have shown there exists a systematic hierarchyof several levels for the local relativistic approximation toHamiltonian matrix elements In this section we study theaccuracy of the different levels on a test molecule set and in-troduce the following notation to distinguish the results

(1) FULL Full molecular relativistic transformation no ap-proximation

(2) DLU Diagonal local approximation to decouplingtransformations ULL and ULS are evaluated for eachatomic (diagonal) block However three variants arepossible(i) DLU(ALL) Calculate relativistic transformation

for all atoms (all atoms are treated as ldquoheavyrdquo el-ements) That is apply the ldquoatomicrdquo unitary trans-formation to the diagonal and to all off-diagonalblocks of the Hamiltonian

(ii) DLU(NB) Calculate relativistic transformation forall atoms (all atoms are treated as ldquoheavyrdquo el-ements) but apply the relativistic transformationonly to blocks of the Hamiltonian assigned toneighboring atoms (and to the diagonal blocks ofcourse) to achieve linear scaling

(iii) DLU(ABCBP) Distinguish heavy and lightatoms where ABC denote the heavy atoms and ap-ply the BP approximation to the light atoms

(iv) DLU(ABCNR) Distinguish heavy and lightatoms where ABC denote the heavy atoms and ap-ply the NR approximation to the light atoms

(3) DLH Diagonal local approximation to Hamiltonianmatrix blocks The relativistic correction for off-diagonal ldquointeratomicrdquo blocks are neglected(i) DLH(ALL) Calculate a relativistic description for

all diagonal ldquoatomicrdquo blocks(ii) DLH(ABCNRBP) Only heavy-atom blocks are

considered for the relativistic description

Note that this classification system does not account for allpossible local relativistic approximations (i) A could also rep-resent a group of atoms and (ii) additional off-diagonal ap-proximations may also be included

We have implemented the local relativistic schemes dis-cussed so far for the X2C BSS and DKH exact-decouplingapproaches into the MOLCAS programme package61 intowhich we have recently implemented X2C BSS and thepolynomial-cost DKH schemes52 In our scalar-relativisticcalculations the definition of the W matrix was

Wij = 1

4c2〈λi | p middot V p|λj 〉 (63)

and λi now refers to scalar basis functions instead of 2-spinorbasis functions All other expressions derived in this workare the same for the scalar-relativistic X2C BSS and DKHvariants We should mention that we have not consideredthe transformation of the two-electron integrals as it iscommon in most exact-decoupling calculations Howeverthe local methods discussed here could be of value also forone of the approximate methods available to transform the



W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352

FIG 1 Structures of the test molecule set (white spheres are hydrogenatoms selected bond lengths are given in Aring)

two-electron integrals (note that a full-fledged transformationof the two-electron integrals would render the application ofexact-decoupling methods inefficient and a four-componentapproach should be employed instead)

For our test molecule set we should rely on closed-shell molecules with heavy elements in the vicinity of otherheavy elements and of light elements Four molecules wereselected to test the validity of the different local schemes I+5 WH6(PH3)3 W(CH3)6 and Pb2+

9 Moreover two reactions

I+5 minusrarr I+3 + I2 (64)

WH6(PH3)3 minusrarr WH6(PH3)2 + PH3 (65)

were chosen to study reaction energies Their structures (seeFig 1) have been optimized with the TURBOMOLE programpackage62 employing the BP86 density functional63 64 with avalence triple-zeta basis set with polarization functions on allatoms65 in combination with Stuttgart effective core poten-tials for W Pb and I as implemented in TURBOMOLE

All molecules are closed shell and for our analysis all-electron HartreendashFock calculations were then performed withthe MOLCAS programme package Since our aim is to test thevalidity of local relativistic approximations instead of obtain-ing accurate results which can be compared to experimentswe do not need to consider electron-correlation effects inour calculations The all-electron atomic natural orbital basissets66ndash68 were employed at a double-zeta level for all atomsThe exact-decoupling DKH calculations were performed at35th order since higher order results did not improve theaccuracy for our purpose so that this high DKH order canbe considered exact Results of finite-order (standard) DKH2calculations are also given for the sake of comparison

The computation times on the Intel i7-870 centralprocessing unit (CPU) for the different local schemes arecompared at the example of Pb2+

9 Table I presents the timingsfor the evaluation of the relativistic one-electron Hamiltonianmatrix operator (The picture change transformation ofproperty integrals is not considered in the measurement ofthe CPU time) For comparison the computation times forthe one-electron and for the electron repulsion integralsare 8 s and 29744 s respectively They are of course thesame for the different relativistic approaches Since therelativistic transformations are performed on each atomic

TABLE I Computation times (in seconds) for the relativistic one-electronHamiltonian in various local schemes (for Pb2+

9 )

DKH35 BSS X2C DKH2

DLH(ALL) 1216 6 6 1DLU(ALL) 1225 15 15 11FULL 124 900 564 520 92

block in the DLH scheme its computational costs aremuch smaller than the FULL scheme The ratio of DLHto FULL is almost a constant (around 1 in this case) forall relativistic approaches The DLU scheme has an extratransformation step for off-diagonal blocks in addition to theDLH scheme it therefore requires a constant time (about10 s in this case) more than the DLH scheme which isstill negligible when compared to the FULL scheme Thecomputation times for the evaluation of both one-electronintegrals and relativistic transformations (except for DKH35)are smaller than that for two-electron integrals The rela-tivistic transformation is usually not the bottleneck stepHowever if density functional theory calculationsmdashwith apure generalized-gradient-approximation functional in com-bination with some acceleration scheme like density fittingor fast multipole approximationsmdashinstead of HartreendashFockcalculations were performed the computation time of therelativistic transformation would be significant and the localapproximation would then be required Of course this alsoholds especially for much larger molecules with light atoms(see also discussion below) For the high-order DKH methodsuch as DKH35 employed here the computational cost of theFULL scheme even exceed that of the two-electron integralsand therefore a local approximation is mandatory

We performed DLU(ALL) and DLH(ALL) calculationsfor all four molecules in order to analyze the general error thatis introduced by these local approximations The differencesof total electronic energies with respect to the FULL referencecalculation without any local approximation are presented inTable II Since the errors of total energies are rather smallall values are given in milli-Hartree (mH) atomic units Theerrors in total energies can be understood as errors on atom-ization energies that one would observe when atomization en-ergies are to be calculated with one of the local approachescompared to the FULL one

For the I+5 molecule which possesses a pseudo-one-dimensional structure with iodine atoms connected to at mosttwo other iodine atoms the DLU(ALL) scheme producesan error of about 0012 mH for both DKH and BSS whileX2C has a slightly larger error of 0013 mH Interestingly itis smallest for DKH2 ie only 0008 mH The DLH(ALL)local scheme produces relatively large errors on the totalelectronic energies they are 024 mH for DKH and BSS 026mH for X2C and 029 mH for DKH2 Clearly DLU(ALL)provides more accurate total energies than DLH(ALL) asit includes the relativistic form of the atomndashother-atom off-diagonal blocks while DLH(ALL) only takes into account theatomndashsame-atom diagonal blocks The errors of DLH(ALL)are about 20 times larger than those of DLU(ALL) for theI+5 molecule This indicates that the relativistic corrections to



TABLE II Total energy differences introduced by local approximations DLU(ALL) and DLH(ALL) with respect to the FULL reference energy All valuesare in milli-Hartree atomic units

Method Approximation I+5 WH6(PH3)3 W(CH3)6 Pb2+9

DKH35 FULL minus 355628190554 minus 171709913602 minus 163752430202 minus 1879059067440DLU(ALL)ndashFULL minus 00119 minus 00126 minus 00204 00376DLH(ALL)ndashFULL 02471 minus 308580 minus 63823 minus 22619

BSS FULL minus 355628315438 minus 171710443875 minus 163752960488 minus 1879072310132DLU(ALL)ndashFULL minus 00119 minus 00125 minus 00204 00387DLH(ALL)ndashFULL 02461 minus 308303 minus 63749 minus 22477

X2C FULL minus 355618410814 minus 171710536272 minus 163753113620 minus 1879112518227DLU(ALL)ndashFULL minus 00133 minus 00306 minus 00367 00523DLH(ALL)ndashFULL 02629 minus 333336 minus 70463 minus 28362

DKH2 FULL minus 355530686409 minus 171587763373 minus 163630319017 minus 1877120963589DLU(ALL)ndashFULL minus 00083 00011 00143 00122DLH(ALL)ndashFULL 02897 minus 334347 minus 69555 minus 33895

off-diagonal blocks are quite important ApparentlyDLU(ALL) provides an accurate scheme for taking theoff-diagonal relativistic description into account

The Pb2+9 molecule features a more compact structure

where more than two heavy atoms bind to a central leadatom Moreover lead atoms are heavier than iodine atomsand relativistic effects are more pronounced As we cansee from Table II errors of Pb2+

9 are all larger than thoseobserved for the I+5 molecule The deviations of DLU(ALL)electronic energies from the FULL reference energy are0038 mH for BSS and DKH while X2C is still slightlylarger with a value of 0052 mH (again DKH2 features thesmallest deviation of only 0012 mH) However deviations ofDLH(ALL) results from the reference have become signifi-cantly larger they are 227 225 and 284 mH for DKH BSSand X2C respectively Hence the errors of DLU(ALL) areroughly 60 times smaller than those observed for DLH(ALL)

Considering now molecules that also contain light atomsthe relative accuracy of DLU(ALL) vs DLH(ALL) changesdramatically First of all we note that our DLU(ALL) schemepreserves its accuracy and the deviations from the FULL ref-erence electronic energy are in between those obtained forI+5 and Pb2+

9 with the same trends as already discussed Theerrors of DLH(ALL) energies obtained for the WH6(PH3)3

molecule are however larger than 30 mH That is theyare significantly larger than those observed for Pb2+

9 ForW(CH3)6 the deviations from the reference are around6 mH which are also still larger than for Pb2+

9 and certainlynon-negligible Note that the DLH(ALL) errors obtained forWH6(PH3)3 and W(CH3)6 are not all below those of the Pb2+

9molecule although the nuclear charge number of tungstenis smaller than that of lead This is in contrast to the re-sults obtained for our DLU(ALL) scheme Since DLH ne-glects any relativistic correction to off-diagonal ldquointeratomicrdquoblocks the large errors must be due to the strong coupling be-tween tungsten and its surrounding lighter atoms (phospho-rus and carbon) Apparently if a heavy-element-containingmolecule has several strong bonds to this heavy element theoff-diagonal blocks of neighboring atom pairs can have sig-nificant contributions to the total electronic energy Thereforethe DLH scheme may then introduce non-negligible errors

Since DLU(ALL) gives a balanced description for allblocks it will still work in those cases where DLH(ALL) failsDLU can therefore even be recommended for calculations ofmolecular systems in different geometries as for example re-quired for the calculation of potential energy surfaces DLHwould be unfit for this case as it will introduce large errorswhen the interatomic distances become short

DLU(ALL) also works well for the finite-order (stan-dard) DKH2 method The errors are even smaller comparedto exact-decoupling approaches However note that the refer-ence energies of the different relativistic approaches are notthe same Since the DLU(ALL) scheme only discards the off-diagonal blocks of the decoupling transformation comparedto the FULL reference scheme the small errors in the to-tal electronic energies indicate that neglecting off-diagonalterms in the decoupling transformation produces negligibleerrors for the total energies This conclusion holds for bothexact-decoupling and finite-order DKH approaches With theDLH(ALL) scheme however the errors in the DKH2 cal-culation that are due to the DLH approximation are of thesame order as those introduced by the finite ie second orderwhen compared to the exact-decoupling approaches This ob-servation leads to the very important conclusion that none ofthe exact-decoupling approaches is superior to a finite-orderDKH approach if one employs a DLH scheme as this wouldrender the accuracy gained by exact decoupling meaninglessIn fact if DLH is for some reason the local approximationof choice a standard DKH2 approach would yield a moreefficient relativistic scheme and nothing would be gained byachieving exact decoupling

Besides total energies we also study the effects of lo-cal approximations on molecular properties We calculatedthe picture change corrected quadrupole moments The prop-erty matrices are transformed according to Eq (41) The val-ues for the components of the quadrupole moment dependon the orientation of the molecule Therefore we presentthe spherically averaged quadrupole moments defined asradic

(Q2XX+Q2

YY+Q2ZZ)3 in Table III where QXX denotes the XX

(diagonal) component of the quadrupole moment The spher-ically averaged quadrupole moments are isotropic and thusindependent of molecular orientation



TABLE III Spherically averaged quadrupole moments for our test set of molecules All values are given inHartree atomic units


DKH35 FULL 8384331 5761285 5729508 17368894DLU(ALL)ndashFULL minus 000009 000018 000025 000416DLH(ALL)ndashFULL 002176 minus 002448 minus 000218 minus 020252

BSS FULL 8384329 5761285 5729507 17368801DLU(ALL)ndashFULL minus 000009 000018 000025 000414DLH(ALL)ndashFULL 002177 minus 002449 minus 000218 minus 020248

X2C FULL 8384406 5761240 5729499 17368292DLU(ALL)ndashFULL minus 000017 000029 000027 000590DLH(ALL)ndashFULL 002199 minus 002483 minus 000197 minus 021653

DKH2 FULL 8385310 5761393 5729674 17380677DLU(ALL)ndashFULL 000009 minus 000012 000018 minus 000003DLH(ALL)ndashFULL 002196 minus 002458 minus 000186 minus 020760

The relative errors of DLU(ALL) are all below 0001this upper limit can be further reduced to 00001 if we donot include Pb2+

9 Hence DLU(ALL) is not only a good ap-proximation for total electronic energies but also for molecu-lar properties such as quadrupole moments The errors intro-duced by DLU(ALL) are all negligible whereas DLH(ALL)shows large errors for quadrupole moments While for to-tal electronic energies the largest error of the DLH(ALL)scheme shows up in WH6(PH3)3 and the smallest in I+5 itis for the quadrupole moments largest for Pb2+

9 and smallestfor W(CH3)6 This different behavior of DLH(ALL) indicatesagain the important contribution of off-diagonal ldquointeratomicrdquoblocks whose importance must be different for electronic en-ergies and for properties In our DLU(ALL) scheme I+5 hasthe smallest error and Pb2+

9 the largest one for both ener-gies and properties Therefore DLU(ALL) turns out to give aconsistent description of relativistic effects while DLH(ALL)yields an unbalanced relativistic treatment We should stressthat we have also investigated other properties (radial mo-ments electric field gradients contact densities) drawing ba-sically the same conclusions (which is the reason why we donot report those data here)

For both energies and quadrupole moments in theDLU(ALL) scheme DKH and BSS yield almost the samedeviations from the reference and X2C always has largerones than DKH and BSS This similarity of DKH and BSScan be well understood since both of them employ the free-particle FW transformation as well as the subsequent exact-decoupling transformation The close results of DKH andBSS were also observed in Ref 52 The slightly larger er-rors of X2C indicate that the off-diagonal blocks of the X2Cexact-decoupling transformation have slightly larger contri-butions compared to DKH and BSS However such differ-ences of errors are too small to be considered a serious draw-back for the X2C approach they are smaller than the dif-ferences of the total values We may use WH6(PH3)3 as anexample The difference of total isotropic quadrupole mo-ments (5761285minus5761240) is 045 milli atomic units whilethe difference of errors by DLU(ALL) (000029minus000018)is 011 milli atomic units The errors introduced by theDLU(ALL) approximation are smaller than the differences

between exact-decoupling approaches This is also the casefor the differences between DKH2 and exact-decoupling ap-proaches As we can see the difference between DKH2and BSS of total isotropic quadrupole moments (5761393ndash5761285) is 108 milli atomic units which are much largerthan the DLU(ALL) errors

For chemical purposes comparisons of total electronicenergies are meaningless because chemical reaction kineticsand thermodynamics are governed by energy differencesWe therefore calculated the electronic reaction energies ofthe two reactions mentioned above The results are given inTables IV and V The errors of relative energies are lessthan those of absolute total energies For example with theDKH approach the error of I+5 is reduced from 0012 mH to0002 mH for DLU(ALL) 024 mH to 012 mH forDLH(ALL) For the WH6(PH3)3 molecule the error isdecreased from 00126 mH to 00006 mH for DLU(ALL)309 mH to 194 mH for DLH(ALL) This also holds forother relativistic approaches

Next we study the neighboring-atomic-block (NB) ap-proximation at the example of the I+5 molecule As shown inFigure 1 it is composed of a chain of iodine atoms and thussuitable for applying the neighboring approximation Onlythe pairs of atoms which are connected with a bond dis-played in Figure 1 are considered as neighbors The neigh-boring groups could also be determined by introducing acut-off distance parameter Then the pair of atoms whichhas shorter distance than the given parameter is counted asa neighboring pair The relativistic transformations are nowonly applied to such neighboring pairs according to Eq (22)

TABLE IV Electronic HartreendashFock reaction energies for the reactionI+5 minusrarr I+3 + I2 All values are in milli-Hartree (note that 1 mH is equivalentto about 26 kJmol)

DKH35 BSS X2C DKH2

FULL 139797 139798 139739 139550DLU(ALL)ndashFULL 00022 00022 00026 00011DLU(NB)ndashFULL 00040 00040 00044 00027DLH(ALL)ndashFULL minus 01190 minus 01190 minus 01214 minus 01231



TABLE V Relative energy difference of the reaction WH6(PH3)3

minusrarr WH6(PH3)2 + PH3 All values are in milli-Hartree

DKH35 BSS X2C DKH2

FULL 358447 358448 358439 359064DLU(ALL)ndashFULL minus 00006 minus 00007 00040 00101DLU(WPNR)ndashFULL minus 01579 minus 01579 minus 01533 minus 01476DLU(WNR)ndashFULL 03648 03648 03684 03737DLU(WPBP)ndashFULL 12501 12500 12547 12608DLU(WBP)ndashFULL 2544914 2544914 2544951 2545018DLH(ALL)ndashFULL 194202 194024 210255 211635DLH(WPNR)ndashFULL 194229 194051 210284 211663DLH(WNR)ndashFULL 196722 196544 212789 214146DLH(WPBP)ndashFULL 203668 203490 219743 221116DLH(WBP)ndashFULL 2730474 2730296 2746570 2747922

within the DLU scheme As we can see from Table IV theneighboring approximation DLU(NB) gives very small er-rors for relative energies Therefore it is a very good (addi-tional) approximation within our DLU scheme Hence withDLU(NB) the computational cost will become linear scalingwhile the DLU(ALL) scheme has an order-N2 scaling whichmay be a bottleneck for calculations on very large moleculesBut be aware that the success of the NB approximation de-pends on how one defines the neighboring atoms If the cut-off distance is too small the DLU scheme will be reduced tothe DLH scheme since no neighbors would be found

Reduction in terms of computational costs can also beachieved by employing the BP or NR approximation to thelight atoms It should be a good approximation if a moleculehas many light atoms such as hydrogen We study these ap-proximations for the phosphine ligand dissociation energyof WH6(PH3)3 The results are presented in Table V whereDLU(WPNR) denotes the NR approximation applied to theH atoms while W and P atoms are relativistically describedBy contrast DLU(WNR) denotes application of the relativis-tic transformation only to the W atom while for P and H theNR approximation applies From Table V we can see that theresults of the BP approximation are very bad especially if theBP approximation was employed for the phosphorus atomsThe BP approximation to P and H atoms gives errors whichare roughly seven times the value for the reactions energyHence this approximation turns out to be completely uselessSuch huge errors stem from the variational collapse of the BPapproximation

However the NR approximation is much better The er-rors on the reaction energy within the DLU(NR) schemewhich are in the range of 01ndash03 mH and thus less than1 kJmol are perfectly acceptable This success is due to thecorrect nonrelativistic limit of the NR approximation withinthe DLU scheme Not unexpectedly the DLH(NR) resultshave quite larger errors because DLH(ALL) already does notprovide good results for the phosphine dissociation energyfrom the WH6(PH3)3 complex If we study the difference be-tween DLH(NR) and DLH(ALL) we find that the NR ap-proximation actually gives small errors on top of DLH(ALL)However the DLH scheme is not recommended since its ac-curacy is not guaranteed at all Only in those cases for whichthe DLH scheme may work we can further add the NR ap-

proximation on top of it for light atoms As can be seen fromboth Tables IV and V different relativistic approaches behavebasically identical with respect to the discussion of the NBand NR approximations Therefore the approximations forthe reduction of computational cost discussed in this papercan be applied to all relativistic transformation approaches

VII CONCLUSIONS

In this work we aimed at a rigorous local approximationto relativistic transformation schemes which makes them ap-plicable to and efficient for calculations on large moleculesWe developed a systematic hierarchy of approximations thatis based on the assembly of the unitary transformation fromldquoatomicrdquo contributions (DLU) rather than on a local approx-imation directly applied to the matrix representation of theHamiltonian (DLH)

The straightforward DLH scheme which has evolved inthe development of the DKH method (see references givenabove) turns out to be not a very accurate local approxi-mation since it only covers the relativistic treatment of theatomndashsame-atom diagonal blocks in the Hamiltonian It mayfail if interatomic distances become short so that a relativistictreatment of the off-diagonal ldquointeratomicrdquo blocks becomesimportant As a consequence the DLH approximation is notsuitable for example in studies that crucially depend on accu-rate electronic energy surfaces Even if the molecular geom-etry is fixed the DLH approach does not provide a balanceddescription for different molecular properties since the con-tributions of the off-diagonal blocks to total observables arequite different for different properties

By contrast the DLU scheme proposed by us in this pa-per overcomes the drawbacks of the DLH scheme It is anexcellent approach to take into account the atomndashother-atom(ie ldquointeratomicrdquo) off-diagonal relativistic transformationsIt does not show an instability for varying molecular struc-tures (ie for electronic energy surfaces) and properties Thesize of the error introduced by DLU is much smaller than thatof DLH it is typically only 1 of the latter The errors of theDLU approach when compared to the full relativistic trans-formation without any local approximation turn out to be tinyThey are even smaller than the difference among the differentexact-decoupling approaches which are already very small

The DLU scheme is valid for all exact-decoupling ap-proaches while the local X-matrix scheme only works forthe X2C approach Furthermore the DLU scheme can alsobe applied to finite-order DKH approaches such as DKH2which has been very successful in computational chemistryand whose computational costs are less than those of anyexact-decoupling approach

If one has to use the DLH scheme for some reasonsuch as the lack of a DLU implementation or because ofits linear-scaling behavior the selection of a relativisticapproaches is not decisive as the errors introduced by theDLH scheme are already higher than the difference ofdifferent relativistic approaches Hence the approximate butfast DKH2 method may be safely combined with the DLHapproximation while this cannot be recommended for anyexact-decoupling approach



However the linear-scaling behavior of DLH is no goodreason since the DLU scheme can also be made linear scalingwithin the neighboring-atomic-blocks approximation whichproduces negligible errors One can further reduce the com-putational costs by employing the BP or NR approximation toall light atoms However the BP approximation turns out tobe not suitable for variational calculations since it introduceslarge errors Only the NR approximation yields acceptable re-sults The errors of the NR approximation will be larger if thenuclear charge of the atom for which the NR approximationis invoked increases Therefore the application of the NR ap-proximation will depend on the balance of accuracy and com-putational cost in an actual calculation

While this work focused on the accurate approximationof exact-decoupling methods at the self-consistent-field levelwith respect to total electronic energies and first-order prop-erties of a small but well-selected test molecule set furtherinvestigations into the DLU scheme concerning for examplehigher order properties spinndashorbit and electron-correlationeffects are under way in our laboratory

ACKNOWLEDGMENTS

This work has been financially supported by the SwissNational Science Foundation

1K Dyall and K Faegri Introduction to Relativistic Quantum Chemistry(Oxford University Press 2007)

2M Reiher and A Wolf Relativistic Quantum Chemistry (Wiley-VCHWeinheim 2009)

3B A Hess Phys Rev A 33 3742 (1986)4G Jansen and B A Hess Phys Rev A 39 6016 (1989)5L L Foldy and S A Wouthuysen Phys Rev 78 29 (1950)6M Douglas and N M Kroll Ann Phys (NY) 82 89 (1974)7T Nakajima and K Hirao Chem Phys Lett 329 511 (2000)8T Nakajima and K Hirao J Chem Phys 113 7786 (2000)9A Wolf M Reiher and B A Hess J Chem Phys 117 9215 (2002)

10C van Wuumlllen J Chem Phys 120 7307 (2004)11M Reiher and A Wolf J Chem Phys 121 2037 (2004)12M Reiher and A Wolf J Chem Phys 121 10945 (2004)13A Wolf and M Reiher J Chem Phys 124 064102 (2006)14A Wolf and M Reiher J Chem Phys 124 064103 (2006)15M Reiher and A Wolf Phys Lett A 360 603 (2007)16D Peng and K Hirao J Chem Phys 130 044102 (2009)17C Chang M Pelissier and P Durand Phys Scr 34 394 (1986)18E van Lenthe E J Baerends and J G Snijders J Chem Phys 99 4597

(1993)19E van Lenthe E J Baerends and J G Snijders J Chem Phys 101 9783

(1994)20J Autschbach S Patchkovskii and B Pritchard J Chem Theory Comput

7 2175 (2011)21F Aquino N Govind and J Autschbach J Chem Theory Comput 7

3278 (2011)22J Autschbach Coord Chem Rev 251 1796 (2007)23M Barysz A J Sadlej and J G Snijders Int J Quantum Chem 65 225

(1997)

24M Barysz and A J Sadlej J Mol Struct THEOCHEM 573 181 (2001)25M Barysz and A J Sadlej J Chem Phys 116 2696 (2002)26D Kedziera and M Barysz Chem Phys Lett 446 176 (2007)27J-L Heully I Lindgren E Lindroth S Lundqvist and A-M

Maringrtensson-Pendrill J Phys B 19 2799 (1986)28K G Dyall J Chem Phys 106 9618 (1997)29K G Dyall J Chem Phys 109 4201 (1998)30K G Dyall and T Enevoldsen J Chem Phys 111 10000 (1999)31K G Dyall J Chem Phys 115 9136 (2001)32K G Dyall J Comput Chem 23 786 (2002)33M Filatov and D Cremer J Chem Phys 119 11526 (2003)34H J A Jensen in REHE 2005 Conference Muumllheim Germany April

200535M Filatov and D Cremer J Chem Phys 122 064104 (2005)36W Kutzelnigg and W Liu J Chem Phys 123 241102 (2005)37W Kutzelnigg and W Liu Mol Phys 104 2225 (2006)38W Liu and D Peng J Chem Phys 125 044102 (2006)39M Filatov and K G Dyall Theor Chem Acc 117 333 (2007)40W Liu and W Kutzelnigg J Chem Phys 126 114107 (2007)41M Iliaš and T Saue J Chem Phys 126 064102 (2007)42D Peng W Liu Y Xiao and L Cheng J Chem Phys 127 104106

(2007)43W Liu and D Peng J Chem Phys 131 031104 (2009)44J Sikkema L Visscher T Saue and M Ilias J Chem Phys 131 124116

(2009)45R Mastalerz G Barone R Lindh and M Reiher J Chem Phys 127

074105 (2007)46B A Hess and C M Marian in Computational Molecular Spectroscopy

edited by P Jensen and P R Bunker (Wiley Chichester 2000) p 16947M Reiher Theor Chem Acc 116 241 (2006)48W Liu Mol Phys 108 1679 (2010)49T Nakajima and K Hirao Chem Rev 112 385 (2012)50M Reiher WIREs Comput Mol Sci 2 139 (2012)51T Saue ChemPhysChem 12 3077 (2011)52D Peng and M Reiher Theor Chem Acc 131 1081 (2012)53C Herrmann M Reiher and B A Hess J Chem Phys 122 034102

(2005)54J E Peralta and G E Scuseria J Chem Phys 120 5875 (2004)55J E Peralta J Uddin and G E Scuseria J Chem Phys 122 084108

(2005)56J Thar and B Kirchner J Chem Phys 130 124103 (2009)57L Gagliardi N C Handy A G Ioannou C-K Skylaris S Spencer

A Willetts and A M Simper Chem Phys Lett 283 187 (1998)58A V Matveev and N Roumlsch J Chem Phys 128 244102 (2008)59E J Baerends W H E Schwarz P Schwerdtfeger and J G Snijders

J Phys B 23 3225 (1990)60R Mastalerz R Lindh and M Reiher Chem Phys Lett 465 157

(2008)61F Aquilante L D Vico N Ferre G Ghigo P-Aring Malmqvist P Neogrady

T B Pedersen M Pitonak M Reiher B O Roos et al J Comput Chem31 224 (2010)

62R Ahlrichs M Baumlr M Haumlser H Horn and C Koumllmel Chem Phys Lett162 165 (1989)

63A D Becke Phys Rev A 38 3098 (1988)64J P Perdew Phys Rev B 33 8822 (1986)65A Schaumlfer C Huber and R Ahlrichs J Chem Phys 100 5829

(1994)66B O Roos R Lindh P-A Malmqvist V Veryazov and P-O Widmark

J Phys Chem A 109 6575 (2005)67B O Roos R Lindh P-A Malmqvist V Veryazov and P-O Widmark

Chem Phys Lett 409 295 (2005)68B O Roos R Lindh P-A Malmqvist V Veryazov P-O Widmark and

A C Borin J Phys Chem A 112 11431 (2008)


THE JOURNAL OF CHEMICAL PHYSICS 136 244108 (2012)

Local relativistic exact decouplingDaoling Penga) and Markus Reihera)

ETH Zurich Laboratorium fuumlr Physikalische Chemie Wolfgang-Pauli-Strasse 10CH-8093 Zurich Switzerland

(Received 15 April 2012 accepted 4 June 2012 published online 27 June 2012)

We present a systematic hierarchy of approximations for local exact decoupling of four-componentquantum chemical Hamiltonians based on the Dirac equation Our ansatz reaches beyond the triviallocal approximation that is based on a unitary transformation of only the atomic block-diagonalpart of the Hamiltonian Systematically off-diagonal Hamiltonian matrix blocks can be subjectedto a unitary transformation to yield relativistically corrected matrix elements The full hierarchy isinvestigated with respect to the accuracy reached for the electronic energy and for selected molecularproperties on a balanced test molecule set that comprises molecules with heavy elements in differentbonding situations Our atomic (local) assembly of the unitary exact-decoupling transformationmdashcalled local approximation to the unitary decoupling transformation (DLU)mdashprovides an excellentlocal approximation for any relativistic exact-decoupling approach Its order-N2 scaling can be fur-ther reduced to linear scaling by employing a neighboring-atomic-blocks approximation ThereforeDLU is an efficient relativistic method well suited for relativistic calculations on large molecules Ifa large molecule contains many light atoms (typically hydrogen atoms) the computational costs canbe further reduced by employing a well-defined nonrelativistic approximation for these light atomswithout significant loss of accuracy We also demonstrate that the standard and straightforwardtransformation of only the atomic block-diagonal entries in the Hamiltonianmdashdenoted diagonal localapproximation to the Hamiltonian (DLH) in this papermdashintroduces an error that is on the order ofthe error of second-order DouglasndashKrollndashHess (ie DKH2) when compared with exact-decouplingresults Hence the local DLH approximation would be pointless in an exact-decoupling frameworkbut can be efficiently employed in combination with the fast to evaluate DKH2 Hamiltonian in orderto speed up calculations for which ultimate accuracy is not the major concern copy 2012 AmericanInstitute of Physics [httpdxdoiorg10106314729788]

I INTRODUCTION

The consideration of relativistic effects is essential tothe proper understanding of the chemistry of heavy-element-containing molecules1 2 The Dirac equation provides therelativistic quantum mechanical description of a single elec-tron in the presence of external electromagnetic potentials Asthe Dirac operator comprises four-dimensional operators itseigenfunctions possess four components The Coulomb inter-action turned out to be sufficiently accurate for the descriptionof chemical phenomena and the corresponding many-electronHamiltonian is called the DiracndashCoulomb Hamiltonian

A relativistic method based on the DiracndashCoulombHamiltonian is called a four-component method It yieldsnegative-energy solutions which are pathologic and of no usein chemistry Also a large number of basis functions are re-quired to properly describe the negative-energy states Theseare two drawbacks of four-component methods that motivatedthe development of two-component methods which removethe negative-energy solutions and can in principle exactlyreproduce the results of four-component calculations2

Several relativistic two-component methods were devel-oped in the past decades One of the widely used approaches

a)Authors to whom correspondence should be addressed Electronicaddresses daolingpengphyschemethzch and markusreiherphyschemethzch

is the second-order DouglasndashKrollndashHess method (DKH2)3 4

It employs the free-particle FoldyndashWouthuysen (fpFW) (Ref5) transformation as well as sequential DouglasndashKroll6 trans-formations to decouple the four-component operator Higherorder7ndash10 and even arbitrary-order11ndash16 DKH methods havebeen developed The zeroth-order regular approximation(ZORA) (Refs 17ndash19) is another highly successful relativis-tic two-component method Within the ZORA framework itis particularly easy to implement the calculation of molecularproperties see Refs 20 and 21 for very recent examples andRef 22 for a review

In recent years it was shown that if the goal is to ex-actly decouple the four-component Hamiltonian matrix in-stead of the Hamiltonian operator the formulas of exactdecoupling can be directly obtained The BaryszndashSadlejndashSnijders (BSS) (Refs 23ndash26) method aims at exact decou-pling of the free-particle FW-transformed four-componentHamiltonian matrix by a matrix operator of the form derivedin Ref 27 The key matrix operator which is used to constructthe decoupling transformation matrix is obtained by solv-ing an iterative equation However invoking the free-particleFW transformation turns out to be not necessary for the con-struction of the exact-decoupling transformation in a one-stepprotocol The pioneering work of this one-step transforma-tion was provided by Dyall28ndash32 in the form of the so-callednormalized elimination of the small component (NESC)

0021-96062012136(24)24410811$3000 copy 2012 American Institute of Physics136 244108-1











Tij = 1

2〈λi | p2|λj 〉 (1)


Vij = 〈λi |V|λj 〉 (2)


Sij = 〈λi |λj 〉 (3)


Wij = 1



H = H (T VW S) (5)







HAB = TAB + VAB (7)




with

HAoplusB =(

HAA HAB

HBA HBB

) (9)








U =(

Z+ 1radic1+XdaggerX

Z+ 1radic1+XdaggerX

Xdagger


1+XXdagger

)equiv

(ULL ULS

USL USS

)

(12)



L

ϕ+S = Xϕ+

L (13)





X = I (15)


Xaa = I (16)













HbdAA = UAADAAU

daggerAA (19)


HbdAB = UAADABU

daggerBB (20)


HbdAB =

sumIJ






HbdAB = UAADABU





H = (ULL ULS

) (V T

T (W minusT )

) (ULLdagger

ULSdagger

) (23)








H NR = T + V (26)


ULLaa = I and ULS

aa = I (27)


Waa = 0 (28)


WAa = 0 (29)










HAA = ULSAATAAU

LLdaggerAA + ULL

AATAAULSdaggerAA

minusULSAATAAU

LSdaggerAA + ULL

AAVAAULLdaggerAA

+ULSAAWAAU

LSdaggerAA (32)

HAB = ULSAATABU

LLdaggerBB + ULL

AATABULSdaggerBB

minusULSAATABU

LSdaggerBB + ULL

AAVABULLdaggerBB

+ULSAAWABU

LSdaggerBB (33)





AAVAa (34)









Qij = 1





A The X2C approach


UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger


1radic1+XXdagger

) (42)


ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL


Xdagger (44)


V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+


X = C+S

(C+

L

)minus1 (46)



ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)




B The BSS approach


T K = SKt (50)




U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)




D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)


U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger


1radic1+RRdagger

) (54)



H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)







(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)






























Tij = 1

2〈λi | p2|λj 〉 (1)


Vij = 〈λi |V|λj 〉 (2)


Sij = 〈λi |λj 〉 (3)


Wij = 1



H = H (T VW S) (5)







HAB = TAB + VAB (7)




with

HAoplusB =(

HAA HAB

HBA HBB

) (9)








U =(

Z+ 1radic1+XdaggerX

Z+ 1radic1+XdaggerX

Xdagger


1+XXdagger

)equiv

(ULL ULS

USL USS

)

(12)



L

ϕ+S = Xϕ+

L (13)





X = I (15)


Xaa = I (16)













HbdAA = UAADAAU

daggerAA (19)


HbdAB = UAADABU

daggerBB (20)


HbdAB =

sumIJ






HbdAB = UAADABU





H = (ULL ULS

) (V T

T (W minusT )

) (ULLdagger

ULSdagger

) (23)








H NR = T + V (26)


ULLaa = I and ULS

aa = I (27)


Waa = 0 (28)


WAa = 0 (29)










HAA = ULSAATAAU

LLdaggerAA + ULL

AATAAULSdaggerAA

minusULSAATAAU

LSdaggerAA + ULL

AAVAAULLdaggerAA

+ULSAAWAAU

LSdaggerAA (32)

HAB = ULSAATABU

LLdaggerBB + ULL

AATABULSdaggerBB

minusULSAATABU

LSdaggerBB + ULL

AAVABULLdaggerBB

+ULSAAWABU

LSdaggerBB (33)





AAVAa (34)









Qij = 1





A The X2C approach


UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger


1radic1+XXdagger

) (42)


ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL


Xdagger (44)


V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+


X = C+S

(C+

L

)minus1 (46)



ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)




B The BSS approach


T K = SKt (50)




U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)




D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)


U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger


1radic1+RRdagger

) (54)



H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)







(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)























HAB = TAB + VAB (7)




with

HAoplusB =(

HAA HAB

HBA HBB

) (9)








U =(

Z+ 1radic1+XdaggerX

Z+ 1radic1+XdaggerX

Xdagger


1+XXdagger

)equiv

(ULL ULS

USL USS

)

(12)



L

ϕ+S = Xϕ+

L (13)





X = I (15)


Xaa = I (16)













HbdAA = UAADAAU

daggerAA (19)


HbdAB = UAADABU

daggerBB (20)


HbdAB =

sumIJ






HbdAB = UAADABU





H = (ULL ULS

) (V T

T (W minusT )

) (ULLdagger

ULSdagger

) (23)








H NR = T + V (26)


ULLaa = I and ULS

aa = I (27)


Waa = 0 (28)


WAa = 0 (29)










HAA = ULSAATAAU

LLdaggerAA + ULL

AATAAULSdaggerAA

minusULSAATAAU

LSdaggerAA + ULL

AAVAAULLdaggerAA

+ULSAAWAAU

LSdaggerAA (32)

HAB = ULSAATABU

LLdaggerBB + ULL

AATABULSdaggerBB

minusULSAATABU

LSdaggerBB + ULL

AAVABULLdaggerBB

+ULSAAWABU

LSdaggerBB (33)





AAVAa (34)









Qij = 1





A The X2C approach


UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger


1radic1+XXdagger

) (42)


ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL


Xdagger (44)


V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+


X = C+S

(C+

L

)minus1 (46)



ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)




B The BSS approach


T K = SKt (50)




U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)




D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)


U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger


1radic1+RRdagger

) (54)



H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)







(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)
























HbdAA = UAADAAU

daggerAA (19)


HbdAB = UAADABU

daggerBB (20)


HbdAB =

sumIJ






HbdAB = UAADABU





H = (ULL ULS

) (V T

T (W minusT )

) (ULLdagger

ULSdagger

) (23)








H NR = T + V (26)


ULLaa = I and ULS

aa = I (27)


Waa = 0 (28)


WAa = 0 (29)










HAA = ULSAATAAU

LLdaggerAA + ULL

AATAAULSdaggerAA

minusULSAATAAU

LSdaggerAA + ULL

AAVAAULLdaggerAA

+ULSAAWAAU

LSdaggerAA (32)

HAB = ULSAATABU

LLdaggerBB + ULL

AATABULSdaggerBB

minusULSAATABU

LSdaggerBB + ULL

AAVABULLdaggerBB

+ULSAAWABU

LSdaggerBB (33)





AAVAa (34)









Qij = 1





A The X2C approach


UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger


1radic1+XXdagger

) (42)


ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL


Xdagger (44)


V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+


X = C+S

(C+

L

)minus1 (46)



ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)




B The BSS approach


T K = SKt (50)




U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)




D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)


U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger


1radic1+RRdagger

) (54)



H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)







(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)
























AAVAa (34)









Qij = 1





A The X2C approach


UX2C =( 1radic

1+XdaggerX1radic

1+XdaggerXXdagger


1radic1+XXdagger

) (42)


ULLX2C = 1radic

1 + XdaggerX (43)

ULSX2C = ULL


Xdagger (44)


V T

T (W minusT )

)(C+

L

C+S

)=

(S 0

0 12c2 T

)(C+

L

C+S

)ε+ (45)

where C+L and C+


X = C+S

(C+

L

)minus1 (46)



ULSX2C = ULL

X2CXdagger (48)

with S defined as

S = S + 1

2c2XdaggerT X (49)




B The BSS approach


T K = SKt (50)




U0 =(

ULLfpFW ULSfpFW

USLfpFW USSfpFW

)=

⎛⎜⎝radic

E0+c2

2E0

radicE0minusc2

2E0f

minusradic

E0minusc2

2E0

radicE0+c2

2E0f

⎞⎟⎠

(52)




D0 = U0

(K 0

0 K

)dagger (V T

T (W minusT )

) (K 0

0 K

)U

dagger0

(53)


U1 =( 1radic

1+RdaggerR1radic

1+RdaggerRRdagger


1radic1+RRdagger

) (54)



H =(

Kminus1 0

0 Kminus1

)dagger

U1D0Udagger1

(Kminus1 0

0 Kminus1

) (55)







(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)


























(56)



(57)


C The DKH approach


U (m) =1prod

k=m

Uk (58)


Uk =[mk]sumi=0

akiWik (59)



UDKH =(

ULLDKH ULS

DKH

USLDKH USS

DKH

) (60)


DKH are


(61)


(62)















Wij = 1





W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)






















W

WWP

PP

P

P

P

C

C

C

C

C

C

I

II

I

I

I

I

I

PbPb

PbPb

Pb

Pb

Pb PbPb

270

270

300

300

271

271

249249

247

247

247

222

352 352












9 )

DKH35 BSS X2C DKH2


























(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)









































(Q2XX+Q2




















DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)





































DKH35 BSS X2C DKH2






DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)
























DKH35 BSS X2C DKH2






VII CONCLUSIONS










ACKNOWLEDGMENTS










(1997)
























ACKNOWLEDGMENTS










(1997)





















local relativistic exact decoupling - eth z...the consideration of relativistic effects is essential...

Documents