Ephraim Eliav

Relativistic multi-root multireferencecoupled cluster method: state of the art

Collaboration:

U. Kaldor, A. Landau, N. Borschevsky, H. Yakobi (Tel-Aviv);

T. Saue (Strasbourg); L. Visscher (Amsterdam) [DIRAC]

Y. Ishikawa (Puerto-Rico) [Atomic code]

S. Pal, K. R. Shamasundar (Pune) [HSCC]

15.07.2008

Plan of the talk

Introduction. Benchmark calculations of heavy open

shell quasi-degenerate atomic and molecular systems.

No-virtual-pairs approximation and beyond.

Multi-root Effective Hamiltonian approach.

Intermediate Hamiltonian methods.

Different multi-root MRCC schemes and selective

atomic applications.

Conclusions

In heavy open shell quasi-degenerate systems

RELATIVISTIC, QED & CORRELATION (dynamic and nondynamic) effects

1. Are often of the same order of magnitude

2. Nonadditive and strongly intertwined

Relativity (+QED) and correlation effects should be treated

1. Simultaneously + on equal footing

2. Up to high orders + size extensively

THE METHOD OF CHOICE for benchmark calculations :

Multi-root multireference coupled cluster, based on:

Present: Dirac-Coulomb-Breit Hamiltonian (NVPA+ QED effects)

Future : A covariant many-body QED approach

• Dirac-Coulomb-Breit Hamiltonian; “no-virtual-pairs-approximation” (NVPA)

NO Retardation and NO virtual pairs. Not covariant, correct to α2; Sucher 1980

where hd – single electronic Dirac Hamiltonian

Λi – projection onto the positive energy-spectrum of hd

V – finite size nuclear potential

Energy independent Breit interaction, 1932-33

• Atoms: QED corrections “on top” NVPA – Lamb energy shift (VP+SE)

Very approximately:

Vacuum polarization (VP) – using Uehling potential

Self energy (SE) – using Mittleman approx. (scaling to hydrogenic-like atom)

• Molecules: NVPA = IOTC (infinite order two-component) method (Barysz, Sadlej,

2002; Kuttzelnigg, Liu 2005; Reiher 2006)

Current framework of relativity treatment

( ) ( )Dh c p V rα β= ⋅ + − +1

( )( )[ ]Br

r r r12

12

1 2 1 12 2 12 12

21

2= − • + • •

α α α α /

( )( ) i (1 )N N

DCB D i j ij ij j i

i i j

H N h r B+ + + + +

<

= + Λ Λ + Λ Λ∑ ∑

Multi-root multi-reference approach

Multi-root : HΨa= EaΨa ; a=1,..,d. d-dimensional target space

Here:

Multi-reference : - model space (≥d- dimensional) ;

Q=1-P – reciprocal space

Effective Hamiltonian:

Wave operator: ΩΨ0a=Ψa BLOCH EQUATION (BE):

Normalization condition

– Intermediate normalization (INN) P=PΩP ⇒

Effective Hamiltonian: Heff=PHΩP;

Bloch Equation : QHΩP=QΩPHΩP

– Isometric normalization (ISN) P=PΩ+ΩP ⇒

Effective Hamiltonian: Heff=P Ω+HΩP

Bloch Equation : QHΩP=QΩ PΩ+ HΩP

eff

H HΩ = Ω

0 0 0 ; , 1, 2,..., ;a a a a a

effH E C a d Pµµ

µ µΨ = Ψ Ψ = = ∈∑

; P P Pµ µ

µ

µ µ= =∑

µµ µ0

00 where, EHVHH =+=

Generic MR - coupled cluster equation

Generic exponential normal order parameterization:

MR-CC equation:

is solved iteratively (Jacobi algorithm):

If EP(0) ≈EQ

(0) – iterations beset with convergence problems (intruder states)

Intruder States Problem Solutions:

1. Incomplete Model Space (IMS) procedures.

2. Regularization of Jacobi algorithm.

3. State Specific (SS) formulations.

4. Brillouin-Wigner multi-root formulation.

5. EOM- like formulation.

6. INTERMEDIATE HAMILTONIAN (IH) approach.

∑=Ωl

lS exp

[ ] toequal is , ,0 PHVQPHSQconnleffl Ω−Ω=

(0) (0)/( )l eff P Ql,connQS P Q VΩ ΩH P E E= − −

Intermediate Hamiltonian (Hi) - a generalization of Heff

and solution of the intruder states problem

Malriue, 1985: HI is defined in P-space (P=P

m+P

i), but in contrast to Heff only

part of the Hi eigenvalues (namely Nm) are eigenvalues of the exact Hamiltonian ⇒ Freedom in defining problematic QSP

iamplitudes.

In order to diminish this freedom and make the approach more general and flexible we also use the splitting Q=Q

m+Q

i⇒ Freedom in defining problematic QiSP

iamplitudes

Possible modeling of eq. (4) for QiSPiand the appropriate IH schemes:

1. Qi ΩPmHΩ Pi= Qi HΩ Pi ⇒ Scheme IH-1 (IH, Malriue, 1985)

2. PT –based; 0- order: QiSPi=0 ⇒ Scheme IH-2 (IMS, Kaldor, 1979)

3. Qi [S,H0+ Pi ∆]Pi = Qi(HΩ - Ω HΩ +β∆S)Pi ⇒ Scheme XIH (IH, Mukherjee,1992)

Successful modeling of QiSP

imakes it possible to avoid intruder states,

while increasing precision of calculations, using much larger Pm

spaces.

[ ]

[ ]

[ ]

[ ]

0 ,

0 ,

0 ,

0

, (1)

, (2)

, (3)

,

m l m m m i ml conn

i l m i m i ml conn

m l i m m i il conn

i l i i

Q S H P Q V P H PH P

Q S H P Q V P H PH P

Q S H P Q V P H PH P

Q S H P Q V

= Ω − Ω − Ω

= Ω − Ω − Ω

= Ω − Ω − Ω

= ,

(4)m i il conn

P H PH PΩ − Ω − Ω

QmE

0 Pi & Qi

Pm

Relativistic multi-root MRCC approaches

implemented within IH in TAU so far…

• Standard:

1. Fock-space or valence universal

2. Hilbert-space or state universal

• Novel:

Mixed-sector

• In progress:

Double Fock-space – a covariant MRCC

Valence Universal (VU) or Fock-Space (FS)

multi-root MRCC method

Fock-space parameterization: (Lindgren (1979))

Start from some reference single closed shell state P (Fermi vacuum).

P could be ionized or excited state of a system under consideration.

(n,m) – Fock-space sector ≡ subspace of Hilbert space with n – holes, m - particles

XIH-FSCC equations in the case of complete model space and INN (PΩP=P):

Solved hierarchically, using SUBSYSTEM EMBEDDING CONDITION (SEC):

The equations for S(k,m) involve only S(i,j) with j≤m, i ≤k ⇒

First, solve the S(0,0) equations; Next, solve for S(0,1) and/or S(1,0);

Continue as needed.

∑∑=Ωmn

mn

l

l

PS,

),(exp

( , )( , )

0 , , ( )

n mn m

l i conn i l connQ S H P P Q V P V SP Pβ + ∆ = Ω − Ω Ω + ∆

Valence Universal Effective Hamiltonian

Diagonalization of the Heff

(k,m)=P (k,m)(HΩ)conn P (k,m) gives directly the transition

energies with respect to closed shell reference.

All transition energies (IP, EE, or EA) are calculated simultaneously.

Size-extensivity is maintained independently for core and valence electrons.

Symmetry and spin adaptation (JJ or LS) is automatic.

Q large and various ⇒ dynamic correlation is described very GOOD

Fock-space approach – a natural framework for developments of the covariant correlation methods based on QED theory. Double FS-CC:

1) treats electrons and uncontracted virtual photons q-mechanically;

2) couples electronic and photonic degrees of freedom (FS sectors);

3) includes QED multi-photon interactions iteratively (to infinite order);

Drawbacks : Heff – has “diagonal” structure (contains only interactions

between states belonging to the same Fock space sector) ⇒⇒⇒⇒ nondynamiccorrelation – relatively POOR ; NO RELAXATION in low lying sectors.

Relativistic FSCC implementation

(0,0) (0,1)(1,0)

(1,1)(2,1) (1,2)

(2,0) (0,2)

(3,0) (0,3)

(4,0) (5,0) (6,0) (0,5) (0,4)(0,6)

1-comp.(DKH-2) CCSDT(Q)-level; the ( & ) sectors implemented

2- and 4-component CCSD-level: the ( ) sectors implemented

Implementation of higher sectors of FS ( ) is in progress

Atoms –radial symmetry; molecules – double point groups symmetries

Benchmark Atomic Applications

He

Li Be B C N O F Ne

Na Mg Al Si P S Cl Ar

K Ca Sc Ti Ni Cu Zn Ga Ge As Se Br Kr

Rb Sr Y Zr Pd Ag Cd In Sn Sb Te I Xe

Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

Fr Ra Ac Rf 105 106 107 108 109 110 111 112 113 114 115 116 117 118

119 120 121 122 123 124 125 126

Ce Pr Nd Sm Eu Gd Tm Yb Lu

Th Pa U Pu Am Cm Md No Lw

•••• Computed elements:

Published

Unpublished

In progress, or planned

Details of atomic applications:

• Calculated properties include excitation energies (IPs, EEs, EAs, fine structure splittings), HPF parameters, PNC effects etc; For super heavy elements - the nature of the ground states.

• Benchmarking:

1. Radial symmetry is used at both SCF and CC stages; makes possible very large basis sets, L up to 6-8. Basis sets : Uncontracted + kinetic balance

2. Correlate many electrons (40-100) for inclusion of core polarization

3. Inclusion of low-order QED and nuclear effects

4. Convergence of dynamic and/or non-dynamic correlation

Selective FSCC applications. EA of alkali atoms

FS scheme: A+1(0,0)→ A (0,1) → A-1(0,2)

Calculation of the U+4 and the U+5 –one of the most challenging

quantum chemical task

a) Hamiltonians: DC and DCBb) Correlation methods:– XIH-FSCC – extrapolated intermediate Hamiltonian Fock-space coupled cluster.

In the FS schemes the U+6 (Rn-like) reference closed shell used as (0,0) sector:

Pm=[5f+6d+7s+7p]Pi=[6f+7f+8f+9f+7d+8d+9d+10d+8s+9s+10s+11s+8p+9p+10p+11p+

+7g+8g+9g+8h+9h]NOTE: here in the (0,1) sector in order to escape intruders belonging to double-excitation manifold we used thesplitting Q=Qm+Qi in addition to the decomposition P=Pm+Pi

– CASPT2- state-specific based on MCSCF on the valence orbitals[5f+6d+7s+7p] ( CAS )

– Basis set: 37s32p24d21f12g10h9i UBS of Ishikawa and Malli.

[ ] [ ]+6 +5 +4U [0,0] U 0,1 U 0,2 → →

2158862151122160902156632129882P3/2

1933901923511929911911711882952P1/2

1414481402111391251449461422062S1/2

100511100107993121058711036192D5/2

91000905628962595309929892D3/2

760975987833822683842F7/2

(EE)

-508183507258--6p65f5/2

(IP)

Experim.XIH-

FSCC

DCB+

XIH-

FSCC

DC+

CASPT2

DCB+

CASPT2

DC+

State

The U+5 energy levels, cm-1

Table II. The U+4

energy levels, cm-1

State DC+

CASPT2

DCB+

CASPT2

DC+

XIH-

FSCC

DCB+

XIH-

FSCC

Experim.

6p65f

2 3H4

(IP)

401337 402654 380220 381074 -

3F2 (EE) 3742 3773 4190 4202 4161

5f2 3

H5 6746 6631 6275 6070 6137

5f2 3

F3 8989 8897 9147 8974 8983

5f2 3

F4 9892 9779 9586 9404 9434

5f2 3

H6 12676 12486 11780 11420 11514

5f2 1

D2 15196 15106 16785 16554 16465

5f2 1

G4 17599 17391 16937 16630 16656

5f2 3

P0 15546 15556 17840 17837 17128

5f2 3

P1 18500 18426 20570 20441 19819

5f2 1

I6 21306 21089 22812 22534 22276

5f2 3

P2

23753 23539 25315 24991 24652

5f2 1

S0

43483 43361 45765 45611 43614

5f6d 3H4 63221 65821 56289 57161 59183

5f6d 3F2 62542 65172 56475 57324 59640

5f6d 3G3 65353 68182 60510 61331 63053

5f6d 1G4 69659 72154 62641 63336 65538

5f6d 3F3 69537 71826 64141 64845 67033

5f6d 3H5 72542 75044 65052 65755 67606

5f7s 3F2 94548 97573 90411 91410 94070

5f7s 3F3 95059 98083 90965 91941 94614

5f7s 3F4 102614 105500 98168 98921 101612

5f7s 1F3 103108 105987 98967 99713 102407

Conclusions

The FSCC within DCB Hamiltonian is much better than the CASPT2 in the case of all the U+5

levels and the 5f2 and 5f6d manifolds of U

+4. The quality of the U

+4 7s5f levels are compatible for

the two methods (the CASPT2 systematically overestimates energy levels for about 3500 cm-1 and

the FSCC underestimates for 2700 cm-1

).

Average errors are (cm-1

):

CASPT2-

DC

CASPT2 -

DCB

XIH-FSCCSD

DC

XIH-FSCCSD

DCB

U+5

All lines 2577 2704

1602 651

U+4

5f2 (including

1S0) 814 825 514 357

5f6d 3647 6024 2824 2100

5f7s 657 3610 3548 2680

all lines 1526 2665 1953 1215

Effect of the Breit is pronounced and non-additive (within CASPT2 Breit is spoiling the results)!

Hilbert space coupled cluster approach

State universal parameterization (Jeziorski and Monkhost, 1981)

Introduces d different Fermi vacuua (d is the number of Pµ) . ∀∀∀∀ Pµ are generally open shell & belong to the same Hilbert space (MCSCF -like)

Excitation operator Sl(µ) is vacuum-dependent

Hilbert space XIH-CC equation in the case of CMS and INN:

Heff is “nondiagonal” : (Heff)µν = P(µ)(HexpS(ν))connP(ν) - µ and ν could belong

to different Fock space sectors ⇒ nodynamic correlation - GOOD & dynamic correlation – relatively POOR

Problems : - Spin contamination & symmetry breaking! ⇒ special care should be taken for symmetry and spin adaptation.

Implementations: 1-comp. (DKH-2) CCSD and up to 2-valence electrons

2 & 4-comp. – CCSDT and up to 6-valence electrons is under construction

∑∑

=Ωµ

µµ

Pe l

lS

)(

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

0

;

[ , ] ( exp( ) ) exp( ) ( exp( ) ) l i conn i conn l

P

Q S H P P Q V S P S P S P V S Pµ µ µ µ µ ν ν µ µ

ν ν µ

β∈ ≠

+ ∆ = + ∆ − ∑

Pilot HSCC atomic application.Be atom excited states (cm-1). Basis set 6s4p3d.

Hilbert- versus Fock-space XIH-CC

Hilbert-space CC Fock-space CC Experiment

(Pm) orbitals

(Pi ) orbitals

(2s)

(2p3s3p)

(2s2p)

(3s3p3d4s4p)

(2s)

(2p3s3p)

(2s2p)

(3s3p3d4s4p)

3P°(2s2p) 22219 21896 21830 22006 21978 1P°(2s2p) 42897 43137 42689 43256 42565 3S (2s3s) 51667 52279 52265 52392 52080 1S (2s3s) 54797 55348 53351 55466 54677 3P (2s3p) 58202 58799 59890 61022 58907 3P°(2p

2) 60301 58728 59703 60307 59693

1P°(2s2p) 57264 60890 60717 59847 60187

Average

error 765 611 1262 728

Novel mixed-sector multireference CC

(MS-MRCC)

Anzatz & CC equations formally - like in the Fock-Space:

Structure of P is similar to Hilbert-space : few sectors, belonging to the same

Hilbert-space are coupled and diagonalized simultaneously.

Opposite to Hilbert-space approach∀∀∀∀ states from P and Q are build from a single close shell Fermi-state, using Fock-Space strategy – a bridge between Fock-space and Hilbert-space CC;

SEC does not work .The maximal valence rank sector and all lower sectors

are solved simultaneously ⇒ RELAXATION in low lying sectors.

Quasi-closed configurations (p1/22, d3/2

4 , f5/26 ) could be used as a reference P.

Symmetry and spin adaptation is automatic.

Full and balance description of the dynamic & nondynamic correlation.

∑=Ωmn

mn

l PS,

),(exp [ ] PVPVQPHSQ

mn

connlconn

mn

l

),(

)(0

),()(, ΩΩ−Ω=

Mixed-sectors CC method –

implementation in Dirac

RELCCSD is very suited for the MS-CC implementation (all sectors are solved simultaneously).

Pilot implementation in DIRAC: for the study of one-particle states, P contains determinants with

one valence-particle (e.g. P(0,1)), and with 2-particle 1-hole determinants (e.g. P

(1,2)), so called

"shake-up" states.

Structure of Heff: ,)2,1()2,1()2,1()1,0()1,0()2,1()1,0()1,0(

PHPPHPPHPPHPH eff Ω+Ω+Ω+Ω=

Heff=

Pilot MSIH-1 Applications

Electronic configuration (n p3) with J=3/2

Calculations of electronic configurations with tree electrons in the high p orbital (n p

3) with J=3/2, states are combination of n(c1 p

21/2 p

13/2+c2 p

11/2 p

23/2),

combination of Pm and Pi.

Example. The ionization potential (IP) excitation energies (EE) of Bi. Bi Level Expt. MSIH FSCCSD

IP (eV) 6p2

1/2 6p3/2 7.2856 7.2841 7.184 6p

21/2 7s1/2 32588 32451 31789

41125 41079 NC 6p2

1/2 7p1/2 3/2 42941 42945 NC

43913 43924 NC

EE (cm-1

)

6p2

1/2 6d3/2 5/2 44816 44833 NC

31s, 28p, 24d, 20f, 17g, 12h, 9i and 6k. Huzinaga's basis set. Model spaces: Pm valence-particle orbitals: 7-8s, 7p1/2, 6-7p3/2, 6d3/2, 6d5/2. 1) Pi valence-hole orbitals: 6s, 6p1/2, 5d. 2) Pi valence-particle orbitals: 5-6s, 5-6p1/2, 5p3/2, 5d, 4f.

• First covariant many-body theory :

Bethe-Salpeter (BS) eqn 1951 : (E − H0 )Ψ = ν(E)Ψ

ν(E) =

standard QED solution: Brillouin-Wigner-PT using S-matrix (or relevant techniques) –

not feasible beyond two photons; not size-extensive; WF not “updated”

• Steps towards merging QED and quantum-chemical MBPT tools:1) Covariant evolution operator (CEO) method (Lindgren, 2001) – a numerical

procedure suitable for merge QED with time-independent MR-MBPT; in the limit

equals to the Bethe-Salpeter theory. Two formulations (ways of photons treatment):

- Hilbert space: BS-Bloch eqn.; ν(E) - irreducible diagrams; CC iterations – reducible

- Fock-space with variable number of uncontracted virtual photons (photonic sectors)

CC is used (twice!) for 1) numerical generation of ν(E); 2) BS-Bloch eqn. solution

2) Renormalized SCF procedure based on shifted QED- vacuum (T. Saue, M. Levy),

which includes low order radiation effects (VP + SE) in the HF procedure.

Steps towards covariant MRCC method.

Extended Fock space versus Hilbert space

treatment of retarded photons

Fock space The state corresponds to the time t : t0<t<t’ ;satisfies the Fock-space-Schrodinger eqn (derived using CEO)

(H0 + V(x))Ψ = EΨ

Perturbation V(x) is given by the energy-independent electron-field interaction density V(x) = − ψ†Iα

µAµψI

Ψ is a vector in the Fock space with variable number of retarded uncontracted virtual photons.

Projection on Hilbert space gives Bethe-Salpeter eqn(E − H0)Ψ = ν(E)Ψ

The potential ν(E) is given by all “contracted” diagrams

ν(E)=

Hilbert Space The main tool for diagrams’ evaluation in quasi-degenerate case both in Fock-space and Hilbert space is CEO

Bloch equations in generalized

Fock- and Hilbert-spaces (Lindgren, 2001)

• Treat electron-photon interactions in generalized Fock space with variable number of electrons and uncontracted retarded photons.

The Bloch eqn in generalized Fock space

[Ω,H0 ] P = VFSΩ − ΩVeff P ; VFS – energy independent potential

VFS = (0)V + (1)V + (2)V + … - in Fock space potential is divided into terms with

different number of uncontracted virtual photons;

Fock-space double parameterization : Ω=exp∑(µ )S(m,n)(k1l1,…,kµ,lµ)

(m,n) – electronic valence sector; (µ) – number of retarded uncontractedphotons; kn and ln – stand for n’s photon energy and momentum.

• Projection on photonic Hilbert space gives Bethe-Salpeter-Bloch eqn

[Ω,H0 ] P = VHS(Heff)Ω − ΩVeff P VHS(Heff)- energy dependent potential

VHS= V12 + Vret (1) + Vret (2) + … in Hilbert space potential is divided to

instantaneous and retarded, corresponding to exchange with different number of

transverse photons (in parentheses) ;

Fock-space procedure. Zero-order Hamiltonian

• In the generalized Fock-space multi–photonic retarded interactions are evaluated numerically (by contraction of photonic uncontracted lines of VFS with those of Ω in all possible ways) during CC iterations.

• Choice of the zero order Hamiltonian:

1. Lindgren (2001) : (0)V = V12 QED(0)-scheme: H0= HSCF(V12 )

2. Generalization (EE): (0)V = V12 + Vret(1)

QED(1)-scheme: H0=HSCF(V12 + Vret(1) ) – if based on QED vacuum (renormalized filled Dirac sea), includes first order radiative corrections

First order vacuum polarization and electronic self-energy can be treated as parts of direct and exchange terms of the SCF used (0)V = V12 + Vret(1) potential, calculated based on QED vacuum:

VP ∈∈∈∈ direct SE ∈∈∈∈ exchange

SCF term SCF term

Double Fock-space coupled-cluster equation

( ) ( , ) ( ) ( , )

0[ , ] ( )n m n m

eff connS H V Hµ µ= Ω − Ω

The Fock-space excitation operator (µ)S(n,m) and resolvent (µ)RQ(n,m) are divided

into components acting in the subspace with no uncontracted photons (µ=0),

with one photon (µ=1), etc., and present the double FS CC equation as

The generalized FS CC equation (in the case when max µ=2) can then be

separated into:

• The hook represents integration over the photon energy k and summation over the

angular momentum l.

• In the case (0)V = V12 + Vret(1) consideration of the single photonic sector (µ=1)

is sufficient to generate most of irreducible multi-photonic interactions

(0) (0) (0) (0) (1) (1) (2) (2) (0)

(1) (1) (1) (0) (0) (1) (1) (2) (2) (1) (1)

(2) (2) (2) (0) (1) (1) (0) (2)

( )

( )

(

Q eff conn

Q eff conn

Q

S R V V V H P

S R V V V V H P

S R V V V

= Ω + Ω + Ω − Ω

= Ω + Ω + Ω + Ω − Ω

= Ω + Ω + Ω −

(2) )eff conn

H PΩ

Summary.

The multi-root MRCC methods are the most powerful state-of-the-art tools of quantum chemistry, supporting size-extensivity and size-consistency. Few relativistic MRCCs are being developed: Fock-space, Hilbert-space, Mixed-sectors. FUTURE plans: Double FS MRCC – a covariant QED-CC,where CC procedure is used for numerical generation of QED potentials.

Combination of MRCC with Intermediate Hamiltonian method yields many profits: The most flexible and general solution of the intruder states problem. The complete model space P allows the use of the simple Intermediate

Normalization. In the XIH method connectivity and size-extensivity is maintained even for incomplete main model subspace Pm.

Large model spaces allow for good description of the wave function and better agreement with experiment than the traditional approach. States not accessible by traditional methods may be calculated (including “shake-ups” and core ionization).

Convergence wrt P can be studied.

Possibility of using of IH-MRCC program as a “black box”