accurate wavefunctions for multi-sorted quantum systems

Accurate wavefunctions for multi-sorted quantum systems

Michael Pak University of Illinois at Urbana-Champaign

Multi-sorted quantum systems

me ~ mp Type I: positronic chemistry, muonized atoms and molecules

me < < mp Type II: “Chemically” non-adiabatic molecular systems

e: electrons p: positron/muon/etc. Np= 1

e: electrons p: quantum nuclei Np= 1

e p n

quantum classical

Ψ re | R( )→Ψ re,Rp | R( )

•  Mean-field (HF) wavefunction: HF energy: •  Configuration interaction (CI) / MCSCF

CI/MCSCF energy: •  Perturbation theory (MP2) Use 2nd-order perturbation theory to calculate electron-electron and electron-proton corrections

“Imitating” Quantum Chemistry: mean-field based approaches

0, :Φ Φe p0 Slater determinantsΨHF (re ,rp ) =Φ0

e (re )Φ0p (rp )

EHF = Φ0e (re )Φ0

p (rp ) H Φ0e (re )Φ0

p (rp )

ΨCI (re ,rp ) = CII 'Φ Ie (re )Φ I '

p (rp )I '

NCIp

∑I

NCIe

∑

EMP2 = EHF + Eee(2) + Eep

(2)

ECI = ΨCI (re ,rp ) H ΨCI (re ,rp )

Testing electron-positron wavefunctions

•  Positron binding energies - no binding predicted for most systems: e.g., Li, Be, Na, Mg atoms, LiH molecule etc. - accounting for the Ps dissociation channel is problematic

•  Positron annihilation rate - direct measure of quality of e/e+ wavefunctions -

•  Good reference points: - experimental: e+ binding energies and annihilation rates - computational: high accuracy results for small few-electron systems (QMC,SVM, SVM+ECG,,etc.)

Ψ(r1, r2, rp ) = A12 Cijklmnr1ir2

jrpkr1p

l r2 pm r12

ne(−α12

v r12−α1pv r1p−α2 p

v r2 p )v∑

i, j,k,l,m,n∑

Ψ(r1, r2, rp ) = A12 exp(a1r1 + c1r1

2

1+ b1r1+a2r2 + c2r2

2

1+ b2r2+aprp + cprp

2

1+ bprp+a12r12 + c12r12

2

1+ b12r12+a1pr1p + c1pr1p

2

1+ b1pr1p+a2 pr2 p + c2 pr2 p

2

1+ b2 pr2 p)

Ψ (r1e ,r2

e ,rp ) → ρ ep r( ) = Ψ (r1e ,r2

e ,rp ) | δ rie − r( )

i=1

N

∑ δ r p − r( ) |Ψ (r1e ,r2

e ,rp ) → λ 2γ =C ρ ep r( )dr∫

Bound Positrons – Annihilation Rate

>Ψ−Ψ<>=Ψ−Ψ<= ∑∑==

|)(||)(ˆ|41

20

1

20

ee N

i

pei

N

i

pei

Sep crScr rrrr δπδπλ

r0 =α! /mec

0)(ˆ)(ˆ Ω=Ψ ++ qapbi

0)(ˆ)(ˆ Ω−+=Ψ ++ kqpckcf

crH if202 4)0,0(ˆ πσ γ →ΨΨ=

))]3()2()3()2()(1()[,,(ˆ),,( 32123321 αββαα −=Ψ=Ψ rrrfAxxxi

032321321321 )],(ˆ),(ˆ),(ˆ),(ˆ)[,(ˆ),,( Ω↑↓−↓↑↑=Ψ +++++∫ papapapapbpppfdpdpdpi

)]()'()['(),,('ˆ2/1ˆ)( 213321321232 pppppppppfdpdpdpdpAH if −−−=ΨΨ= ∫∑ δδχσχλχ

γ

021 ),(ˆ),(ˆ),'(ˆ)'(' Ω−+↑=Ψ ∫ +++ ekqpcekcpapdpf χ

∫ ===

rrrrrdr

21),(2/1 212 ρσλ γ

q

p

∑=

ʹ′ʹ′=N

iii

HF Scr1

11202πλ

rrrrr dS jijijiij ∫ ʹ′ʹ′ʹ′ʹ′ = )()()()( φφφφ

∑ʹ′

ʹ′ʹ′ʹ′ʹ′ −−+

ʹ′ʹ′−=

rarrar

rra

Srra

cr 11

20

)1( 14

εεεεπλ

λFCI = 2πr02c

CI !I CI !JI !J∑

I !I∑ SIiIi !I p !Jp

i=1

N

∑ +

2 CI !I CJ !J SIiJ j !I p !JpI !I <J !J∑

#

$

%%%%

&

'

(((( 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

[6s]

[6s1p

][6s

2p]

[6s3p

]

[6s2p

1d]

[6s3p

1d]

Basis Setи

(ns-

1)

HF

FCI

MP2

HF

MP2

FCI

6s 6s1p 6s2p1d 6s3p1d 0.2972 0.2976 0.2978 0.2977 0.3490 0.4018 0.5287 0.5376 0.3723 0.6304 0.8662 0.8993

HFλ2MPλ

FCIλ

PsH Annihilation Rate

Dynamical correlation •  Electron-electron dynamical correlation: - defined as difference between exact and mean-field answers E.g. for the energy: - mathematically is (mostly) due to deviation of from when - typically is ‘icing on the cake’: e-e Coulomb interaction is repulsive… •  Electron-proton correlation is the cake! - qualitatively important: e-p Coulomb interaction is attractive

Ecorr = Ψexact H Ψexact − ΨHF H ΨHF

Ψexact ΨHF | rie − r j

e | → 0

ΨXCHF xe ,xp( ) =Φe xe( )Φp xp( ) 1+ g rie ,r j

p( )j=1

Np

∑i=1

Ne

∑$%&

'&

()&

*&

•  Gaussian-type geminals for electron-positron correlation •  bk and γk are constants pre-determined from models •  Variational method: minimize total energy wrt molecular orbital coefficients → Modified Hartree-Fock equations, solve iteratively to self-consistency

g rie ,r j

p( ) = bk exp −γ k rie − r j

p 2"#$

%&'k=1

Ngem

∑Gaussian-type geminals:

Explicitly correlated wavefunctions

ΨRXCHF-fe xe ,x p( ) =χ p x p( )N !

χ1e x1

e( )g r1e ,r p( ) χ2e x1

e( ) ! χNe x1

e( )χ1e x2

e( )g r2e ,r p( ) χ2e x2

e( ) ! χNe x2

e( ). ! " !

χ1e xN

e( )g rNe ,r p( ) χ2e xN

e( ) ! χNe xN

e( )

Method E λ2γ(ns-1)

HF -0.664337 0.3189

XCHF-1G -0.693216 0.6883

XCHF-2G -0.705863 1.1402

XCHF-3G -0.712496 2.0122

XCHF-4G* 0.716097 2.0443

FCI -0.758965 0.8993

SVM -0.789198 2.4714

PsH Annihilation Rate from NEO XCHF

Basis sets: HF. XCHF: 6s FCI: 6s2p1d

2

, 01

( , ) | ( ) ( , )e

e p

N

ep e p x x x e i p e p ei

x x dx dλ ρ δ= ==

= − Ψ =∑∫ ∫ r r r r r:

1 2 31

( , ) ( , , ) ( , , , )e e eN N N

i p i j p i j k pi i j i j k

x x x x x x x x x= ≠ ≠ ≠

Ψ Ω Ψ + Ψ Ω Ψ + Ψ Ω Ψ∑ ∑ ∑2

1( , ) 2i p ip ip ip ip ipx x g gδ δ δΩ = + +

3( , , , )i j k p ip jp kpx x x x g g δΩ =

Ω2 (xi ,x j ,xp ) = 2gipδ jp + gip2δ jp

+2gipg jpδ jp

( ) ( ) ( ) ( )gem 2

, ,1 1 1

, 1 exppe N NN

e p e e p p k k e i p ji j k

b γ= = =

⎧ ⎫⎪ ⎪⎡ ⎤Ψ =Ψ Ψ + − −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭∑∑∑r r r r r r

Two-Photon Annihilation Rates

•  HF very inaccurate •  XCHF better but still inaccurate •  RXCHF-ne/ae perform similarly and agree well with highly accurate ECG/SVM results •  Electronic and positronic densities are also in decent agreement •  RXCHF is 235 (25) times faster than XCHF for e+Li (LiPs)

aMitroy, Phys. Rev. A 70, 024502 (2004). bRyzhikh, Mitroy, Varga, J. Phys. B 31, 3965 (1998).

cStrasburger, J. Chem. Phys. 111, 10555 (1999).

e+Li LiPs e+LiH NEO-HF 3.3556×10-4 0.0512 0.03 XCHF 1.7361 0.566 0.62

RXCHF-ne 1.6759 1.940 1.08 RXCHF-ae 1.6657 1.940 1.11 SVM/ECG 1.7512a 2.107b 1.26c

Units: ns-1

Conclusions: Electron-positron wavefunctions

•  No usable method for computing accurate wavefunctions is available: - highly accurate methods of ECG+SVM type are not practical for systems with more than 5-10 electrons. ((NEO-XCHF/RXCHF is perhaps in the same class)) - methods easily applied to larger positronic molecules (HF, MP2, CI) are utterly unreliable even for systems with 3 electrons or less. - trivial example: experimentally, e+ binds to propane. No known theoretical method can predict this binding… •  The field is wide open, and in need of new ideas…

•  PET applications

Multi-sorted quantum systems: electron-nuclear wavefunctions

•  Solution of mixed nuclear-electronic time-independent Schrödinger equation with molecular orbital methods

•  Treat specified nuclei quantum mechanically on same level as electrons - treat only key H nuclei QM - retain at least two classical nuclei •  Easy access to ‘exact’ answer is available for benchmarking electron-

nuclear orbital methods p :r quantum proton

c :r all other nuclei

Electron-proton wavefunctions: how accurate is BO approximation?

Ψktot re,rp | R( ) = ciµk Ψi

elec re | rp,R( )Ψiµprot rp | R( )

i,µ∑

Hiµ, jν = δijδµνεµi( ) R( ) −

!2

mpΨiµprot | dij

ep( ) ⋅ ∇rp | Ψ jνprot

p−!2

2mpΨiµprot | gij

ep( ) | Ψ jνprot

p

dijep( ) = Ψi

elec | ∇rp | Ψ jelec

egijep( ) = Ψi

elec | ∇rp2 | Ψ j

elece

•  Expand exact solution in double-adiabatic basis

•  Need to evaluate non-adiabatic coupling terms:

Phenoxyl-phenol: electronically non-adiabatic

Benzyl-toluene: electronically adiabatic

C000 = .9973

C000 = .9997

-2 -1 1 2

0.2

0.4

0.6

0.8

1.0

1.2

HF

Electron-proton wavefunctions: what is ‘accurate’?

exact

•  Isotope Effects on Geometries

X HX DX TX

F 0.9186 0.9134 0.9110

Cl 1.2884 1.2815 1.2785

Br 1.4235 1.4165 1.4134

Bond lengths in Å

• NEO-HF → As mass increases, - bond length decreases - magnitude of negative partial charge on X decreases •  Same trends observed for H2O, NH3, H3O+

•  Proton density distribution:

typically, HF frequency is off by 200% or more

He He

Nuclear Quantum Effects and H-bonding •  Large amplitude, anharmonic zero-point motion

Prism Cage

H5O2+ Hydrogen fluoride

•  Clusters: increase H-bond donor–acceptor distances - (H2O)n=2-6 → QDMC: Clary, CR 2000 - (HF)2 → Experiment: Klemperer, JCP 1984 •  Liquids: decrease H-bond donor–acceptor distances - H2O, HF → PICPMD: Klein, JACS 2003, PRL 2004

•  Zero point energy effects: – bending-type modes → increase D–A distance – stretching modes → decrease D–A distance

Proton-Containing Systems with RXCHF

•  14-electron system •  Explicitly correlate two electronic

orbitals to proton (CH bond orbital) Frequency (cm-1)

HF 5077 NEO-RXCHF-ne 3604 NEO-RXCHF-ae 3476 3D grid 3544 black: grid (adiabatic)

blue: NEO-HF red: RXCHF-ne green: RXCHF-ae

N ≡ C – H

F – H – F-

Frequency (cm-1) NEO-HF 4614 NEO-RXCHF-ne 3348 NEO-RXCHF-ae 2616 3D grid 2639

•  20-electron system •  Explicitly correlate four electronic

orbitals to proton (FH bond orbitals)

Problems with explicitly correlated methods for BO systems

ϕi (r){ }i∈ℜ ∈ L2[r] ϕ(r,R)∈ L2[r⊗ R]

0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

electron density: BO(solid line) vs. RXCHF (dotted line)

),( 21 rrΨφ(r1)χ (r2 ) | r1 − r2 |>> 0

Ce−γ |r1−r2| | r1 − r2 |→ 0Φe xe( )Φp xp( ) 1+ g ri

e ,r jp( )

j=1

Np

∑i=1

Ne

∑#$%

&%

'(%

)%

HR[x]= T[x]+ V[x]+1

x − RΨR (x)

Description of Bilobal Wavefunctions

HF: variational solution is always localized In practice, so is CI/CASSCF Delocalization: at least FCI in a very large basis set

Pak and SHS, PRL 2004

D A-H D- A H

• TS for H transfer reactions corresponds to equal H probability near donor and acceptor

Model system studies (analytical proof)

Conclusions •  Electron-positron correlation is too large to make HF a viable reference. •  Post-HF (CI,MP2) methods fail to produce reasonably accurate electron-positron wavefunctions. •  Explicitly correlated methods do somewhat better, but are very expensive.

•  New approaches are needed to describe positron behavior in large molecules.

•  BO ground state electron-nuclear wavefunctions are almost exact even for chemically non-adiabatic systems.

•  Orbital-based methods fail to reproduce the shape and size of exact nuclear density. This is true, to some extent, even for explicitly-correlated approaches.

•  This basic failure probably implies that the functional dependence of electronic wavefunction on nuclear coordinate is described incorrectly.

•  Producing bilobal proton wavefunctions (in a double-well potential) with nuclear-electron orbital methods is essentially an unsolved problem.

Acknowledgments Sharon Hammes-Schiffer (PSU, UIUC) Simon Webb Andrès Reyes Arindam Chakraborty Kurt Brorsen Tzvetelin Iordanov Chet Swalina Jonathan Skone Andrew Sirjoosingh Tanner Culpitt Funding: AFOSR, NSF NEO method is incorporated into GAMESS

Extra: Topology of nodal surfaces

•  Diffusion MC (DMC) depends on the fixed-node approximation. •  Topology of the nodal surface of multi-sorted wavefunctions responds to the level of correlation in an unknown way. •  Nodal surface of an N+M-particle system is an embedded manifold (in R3N+3M). •  Alternatively, after a polynomial approximation, it is an algebraic

variety over R.

•  We need a computable pathway from the embedding function to some suitable topological characteristic (e.g. Betti numbers for some cohomology theory) of the surface.

•  Example: de Rham cohomology groups for (a complement to) an affine variety over C can be computed from the polynomial coefficients of the variety.

Toshinori Oaku, Nobuki Takayama, Journal of Pure and Applied Algebra, 139 (1999) 201

Multicomponent DFT

Parr, JCP1982

Hohenberg-Kohn theorem has been proven for multi-component systems: functional exists for electronic and nuclear 1-particle densities

Kreibich, Gross, PRL 2001

)(),( rr Ne ρρ )(rVext

)(rVext),...,(),( 1 Ne RRr Γρ

Strategy for Developing e-p Functional

•  Geminal ansatz defines map from auxiliary to geminal densities •  Use this map to obtain a functional relationship between 1- and 2-particle geminal densities

Ψgem = (1+G)ΦeΦp!

ep e p, ,ρ ρ ρ !ρ e , !ρ p ,...auxiliary densities geminal densities

ρ ep[ !ρ e , !ρ p ,...]ρ e[ !ρ e , !ρ p ,...]ρ p[ !ρ e , !ρ p ,...]

ρ ep = F[ρ e ,ρ p ]

•  Obtain geminal densities by integration of geminal wavefunction •  Truncate expressions for geminal densities •  Make a well-defined approximation that satisfies sum rules

?

Chakraborty, Pak, SHS, PRL 2008

Electron-Proton Functional

gem 2e p e p

1( , ) exp

N

k kkbg γ

=

⎡ ⎤−⎢ ⎥⎣ ⎦= −∑r r r rGeminal function:

Expression for ep pair density in terms of 1-particle densities:

Contribution to total energy:

e p e p 1 ep e p 1 e pepc ep ep,E d d dr d rρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r r r r

Motivating work: Colle and Salvetti, 1975, developed an electronic functional

e p e p 1 1 p

ep e p e p e p 1 1e p ep

e p e 1 e p p 1e e

ee p p e

1 1 e pe p e

p

p

p

1

gN N

g N N g gN N g

gN gN

ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ

− −

− −

− −

− −

= +

+ 〈〉〈〉+

+ 〈

〈〉

〈〈− 〉 − 〉

〉

( ) egem

p1 G+ ΦΨ Φ=

NEO-DFT for [He-H-He]+

He nuclei classical, H nucleus quantum Electronic basis set: STO-2G Nuclear basis set: 5s

•  NEO-DFT agrees well with NEO-XCHF and grid method •  NEO-DFT ~1400 times faster than NEO-XCHF for this system

He He

Frequencies determined by fitting nuclear density along He-He axis to Gaussian

Isotope NEO-HF NEO-XCHF 3D Grid NEO-DFT H 3759 1030 1107 1072 D 2738 725 783 770 T 2274 558 639 630

NEO-HF

NEO-DFT

3D Grid

Frequencies in cm−1 with H, D, and T for the central nucleus

XCHF vs. RXCHF: Physical Assumptions •  XCHF correlates all electrons with positron using same geminal parameters → geminal functions are used to account for interactions other than short-ranged electron-positron dynamical correlation → wavefunction is not optimized to accurately describe this specific interaction but rather is more globally optimized •  RXCHF correlates only one electron with positron using geminal parameters optimized for a single e-/e+ interaction → geminal functions are used to account mainly for the short-ranged electron-positron dynamical correlation → wavefunction produces highly accurate description of this interaction, although overall wavefunction is not optimal

NEO-DFT for [He-H-He]+

Nuclear basis set: 1s exponent optimized variationally 2 Gaussian-type geminals, scaled with single scaling factor to fit H cc-pVDZ result

•  NEO-DFT provides accurate nuclear densities at low cost! •  Electron-electron and electron-proton correlation predominantly additive

He He

Isotope NEO-HF cc-pVDZ

NEO-DFT cc-pVDZ

3D Grid cc-pVDZ

NEO-DFT cc-pVTZ

3D Grid cc-pVTZ

H 3098 1191 1191 1103 1111 D 2284 820 801 782 740 T 1903 660 633 646 581

Sirjoosingh, Pak, SHS, JCTC 2011; JCP 2012

Frequencies in cm−1 with H, D, T for central nucleus

•  He nuclei classical (fixed) •  H quantum mechanical

PCET: Multiconfigurational RXCHF•  Two-configuration SCF approach for O-H bond on left or right •  Only explicitly correlate 2 electronic orbitals to proton orbital (O-H bond) •  Restricted basis set: only expand explicitly correlated orbitals in terms of valence basis functions on donor, acceptor, and H •  Includes static and dynamical electron-proton correlation •  Substantial computational savings à allows applications to larger systems

I I RXCHF I RXCHF1 1 2 2

II II RXCHF II RXCHF1 1 2 2

C CC C

Ψ = Ψ + Ψ

Ψ = Ψ + Ψ

State 1 State 2

Electronic and Positronic Densities

RXCHF-ne and RXCHF-ae perform similarly and agree well with highly accurate SVM/ECG results

LiPs e+LiH

electronic density positronic density solid: RXCHF-ae dashed: SVM/ECG

SVM data (dashed lines): Ryzhikh, Mitroy, Varga, J. Phys. B 31, 3965 (1998). ECG data (dashed lines): Strasburger, J. Chem. Phys. 111, 10555 (1999).

e+Li

Multicomponent DFT

Strategy for designing electron-proton functionals: •  Define e-p functional as

•  Use an explicitly correlated wavefunction to obtain expression for electron-proton pair density ρ ep(re,rp) in terms of one-particle electron and proton densities ρ e(re) and ρ p(rp) Advantages: provides accurate nuclear densities, consistent treatment of el-el & el-proton correlation Disadvantages: inherent DFT approximations, still expensive

e p e p 1 ep e p 1 e pe

e p e ppc ep ep, ( , ) ( ) ( )rE d rd d dρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r rr rr rr r

Chakraborty, Pak, SHS, PRL 2008; Sirjoosingh, Pak, SHS, JCTC 2011; JCP 2012

e pe

e p e p e pext ref

e pe pcxc pxc

[ , ] [ , ] [ , ]

[ ,[ ] [ ] ]

E E EE E E

ρ ρ ρ ρ

ρ

ρ ρ

ρ ρ ρ

= +

+ + +

Parr, JCP1982; Kreibich, Gross, PRL 2001; Chakraborty, Pak, SHS, JCP 2009

kinetic energy of noninteracting system, “classical” Coulomb

Electron-Proton Functional

gem 2e p e p

1( , ) exp

N

k kkbg γ

=

⎡ ⎤−⎢ ⎥⎣ ⎦= −∑r r r rGeminal function:

Expression for ep pair density in terms of 1-particle densities:

Contribution to total energy:

e p e p 1 ep e p 1 e pepc ep ep,E d d dr d rρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r r r r

Satisfies sum rules:

( ) ( )e e 1 ep e pp p

,Nρ ρ−=r r r ( ) ( )p p 1 ep e pe e

,Nρ ρ−=r r r

Motivating work: Colle and Salvetti, 1975, developed an electronic functional

e p e p 1 1 p

ep e p e p e p 1 1e p ep

e p e 1 e p p 1e e

ee p p e

1 1 e pe p e

p

p

p

1

gN N

g N N g gN N g

gN gN

ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ

− −

− −

− −

− −

= +

+ 〈〉〈〉+

+ 〈

〈〉

〈〈− 〉 − 〉

〉( ) e

gemp1 G+ ΦΨ Φ=

Born-Oppenheimer Approximation

•  Schrödinger equation for electrons and protons •  Fix proton position and solve electronic part •  For each electronic state, solve proton part •  Born-Oppenheimer approximation

Tp +Te +V re,rp,R( )( )Ψktot re,rp;R( ) = Ek R( )Ψk

tot re,rp;R( )

Te +V re,rp,R( )( )Ψielec re;rp,R( ) = εi rp,R( )Ψi

elec re;rp,R( )

Tp + εi rp,R( )( )Ψiµprot rp;R( ) = εµi( ) R( )Ψiµ

prot rp;R( )

Ψktot re,rp;R( ) ≈ Ψi

elec re;rp,R( )Ψiµprot rp;R( )

re,rp,R: electrons, protons, and other nuclei e p n

quantum classical

Non-Born-Oppenheimer Effects

•  Expand exact solution in double-adiabatic basis

•  Express Hamiltonian in matrix form

•  Nonadiabatic couplings provide a measure of nonadiabaticity

Ψktot re,rp;R( ) = ciµk Ψi

elec re;rp,R( )Ψiµprot rp;R( )

i,µ∑

Hiµ, jν = δijδµνεµi( ) R( ) −

!2

mpΨiµprot | dij

ep( ) ⋅ ∇rp | Ψ jνprot

p−!2

2mpΨiµprot | gij

ep( ) | Ψ jνprot

p

dijep( ) = Ψi

elec | ∇rp | Ψ jelec

egijep( ) = Ψi

elec | ∇rp2 | Ψ j

elece

e p n

quantum classical

ep nad

Nuclear Quantum Effects

Zero point energy Vibrationally excited states

Hydrogen bonding Hydrogen tunneling

Proton-coupled electron transfer in solution, proteins, electrochemistry

ET

PT

Nuclear Quantum vs. Non-Adiabatic Effects

)()|(),( RRrRr ΧΦ=Ψ

),(ˆˆˆ),(ˆ RrVHHRrH rR ++=

),(),(),(ˆ RrERrRrH Ψ=Ψ

0),(ˆ ≈Φ RrTR

)|()()|()],(ˆˆ[ RrRWRrRrVHr Φ=Φ+

)()()](ˆˆ[ RERRWHR Χ=Χ+

0),(ˆ ≈Φ RrTR0),(ˆ1

≠Φ RrTR

)|,()()|,()],(ˆ),(ˆˆˆ[ 1111RRrRWRRrRrVRrVHH Rr Φ=Φ+++

)()()](ˆˆ[ RERRWHR Χ=Χ+

)()|,(),( 1 RRRrRr ΧΦ=Ψ

accurate wavefunctions for multi-sorted quantum systems

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