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Anisotropic electron-pair densities in QMC
Manolo Per
RMIT University, Melbourne
August 1, 2008
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 1 / 41
1 Introduction
2 ZVZB
3 Anisotropic Intracule and Extracule Densities
4 Correlation in Stretched Molecules
5 Bonding
6 Summary
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 2 / 41
Definitions
P(r1, r2) =N(N −1)
2
Z
Ψ2(r1, r2, r3, . . . , rN)dr3, . . . ,drN (1)
Full electron-pair density P(r1, r2) is a 6D quantity. Hard to visualize andinterpret.
3D reductions to Intracule density I(u) and Extracule density E(R) retain2-particle information.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 3 / 41
Definitions
Intracule refers to relative separation : rij = ri − rj .
Extracule refers to centre-of-mass : Rij = 12(ri + rj).
I(u) =1
2 ∑i 6=j
〈δ(rij −u)〉
E(R) =1
2 ∑i 6=j
〈δ(Rij −R)〉(2)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 4 / 41
Intracules and Extracules
Investigate electronic structure using more info than density, withoutresorting to orbitals.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 5 / 41
Intracules and Extracules
Investigate electronic structure using more info than density, withoutresorting to orbitals.
Coulson defined the Coulomb hole: Iexact − IUHF . Gives information onelectron correlation effects.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 5 / 41
Intracules and Extracules
Investigate electronic structure using more info than density, withoutresorting to orbitals.
Coulson defined the Coulomb hole: Iexact − IUHF . Gives information onelectron correlation effects.
Ab initio codes: QChem. Lots of four-centre integrals contracted withdensity matrices.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 5 / 41
Intracules and Extracules
Investigate electronic structure using more info than density, withoutresorting to orbitals.
Coulson defined the Coulomb hole: Iexact − IUHF . Gives information onelectron correlation effects.
Ab initio codes: QChem. Lots of four-centre integrals contracted withdensity matrices.
Requires accurate wavefunctions (e.g. CI). QMC may be preferable forlarger systems.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 5 / 41
Intracules and Extracules
Investigate electronic structure using more info than density, withoutresorting to orbitals.
Coulson defined the Coulomb hole: Iexact − IUHF . Gives information onelectron correlation effects.
Ab initio codes: QChem. Lots of four-centre integrals contracted withdensity matrices.
Requires accurate wavefunctions (e.g. CI). QMC may be preferable forlarger systems.
Coulomb holes in DFT - Work of Gori-Giorgi and coworkers.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 5 / 41
Evaluating Densities
Densities in QMC usually evaluated using binning techniques.
Smaller bins lead to larger variances.
Regions of low density have large errors.
Not smooth.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 6 / 41
QMC Work on Densities
Assaraf, Caffarel and Scemema 1 developed efficient estimator for theone-body density,
n(r) = ∑i
〈δ(r− ri)〉 (3)
1R.Assaraf, M.Caffarel, A.Scemema, PRE 75 035701 (2007)2J.Toulouse,R.Assaraf, C.Umrigar, J. Chem. Phys 126 244112 (2007)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 7 / 41
QMC Work on Densities
Assaraf, Caffarel and Scemema 1 developed efficient estimator for theone-body density,
n(r) = ∑i
〈δ(r− ri)〉 (3)
Toulouse, Assaraf and Umrigar 2 developed efficient estimators forspherically averaged Intracule density,
I(u) = (4π)−1Z
I(u)dΩ (4)
1R.Assaraf, M.Caffarel, A.Scemema, PRE 75 035701 (2007)2J.Toulouse,R.Assaraf, C.Umrigar, J. Chem. Phys 126 244112 (2007)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 7 / 41
QMC Work on Densities
Assaraf, Caffarel and Scemema 1 developed efficient estimator for theone-body density,
n(r) = ∑i
〈δ(r− ri)〉 (3)
Toulouse, Assaraf and Umrigar 2 developed efficient estimators forspherically averaged Intracule density,
I(u) = (4π)−1Z
I(u)dΩ (4)
Benefit of low-variance estimators in QMC is the ability to calculate smalldifferences accurately - Correlation effects in vdW systems accessible.
1R.Assaraf, M.Caffarel, A.Scemema, PRE 75 035701 (2007)2J.Toulouse,R.Assaraf, C.Umrigar, J. Chem. Phys 126 244112 (2007)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 7 / 41
QMC Work on Densities
Assaraf, Caffarel and Scemema 1 developed efficient estimator for theone-body density,
n(r) = ∑i
〈δ(r− ri)〉 (3)
Toulouse, Assaraf and Umrigar 2 developed efficient estimators forspherically averaged Intracule density,
I(u) = (4π)−1Z
I(u)dΩ (4)
Benefit of low-variance estimators in QMC is the ability to calculate smalldifferences accurately - Correlation effects in vdW systems accessible.
Key to both (sort of) is the idea of Zero-Variance Zero-Bias.
1R.Assaraf, M.Caffarel, A.Scemema, PRE 75 035701 (2007)2J.Toulouse,R.Assaraf, C.Umrigar, J. Chem. Phys 126 244112 (2007)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 7 / 41
Zero-Variance, Zero-Bias Estimators
R.Assaraf and M.Caffarel, J. Chem. Phys 119 10536 (2003).To estimate 〈O〉:
Define Hamiltonian Hλ = H + λO, and exact wavefunctionΨλ
0 = Ψ0 + λΨ′0 so exact energy is
Eλ0 =
⟨
EλL,0
⟩
Ψλ20
=〈Ψλ
0|Hλ|Ψλ
0〉
〈Ψλ0 |Ψ
λ0〉
(5)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 8 / 41
Zero-Variance, Zero-Bias Estimators
R.Assaraf and M.Caffarel, J. Chem. Phys 119 10536 (2003).To estimate 〈O〉:
Define Hamiltonian Hλ = H + λO, and exact wavefunctionΨλ
0 = Ψ0 + λΨ′0 so exact energy is
Eλ0 =
⟨
EλL,0
⟩
Ψλ20
=〈Ψλ
0|Hλ|Ψλ
0〉
〈Ψλ0 |Ψ
λ0〉
(5)
By Hellmann-Feynman, exact O0 = dEλ0 /dλ|λ=0
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 8 / 41
Zero-Variance, Zero-Bias Estimators
R.Assaraf and M.Caffarel, J. Chem. Phys 119 10536 (2003).To estimate 〈O〉:
Define Hamiltonian Hλ = H + λO, and exact wavefunctionΨλ
0 = Ψ0 + λΨ′0 so exact energy is
Eλ0 =
⟨
EλL,0
⟩
Ψλ20
=〈Ψλ
0|Hλ|Ψλ
0〉
〈Ψλ0 |Ψ
λ0〉
(5)
By Hellmann-Feynman, exact O0 = dEλ0 /dλ|λ=0
Follow same procedure for approximate Ψλ = Ψ+λΨ′, obtain
dEλ/dλ|λ=0 = 〈OL〉Ψ2 +
⟨(
HΨ′
Ψ′−EL
)
Ψ′
Ψ
⟩
Ψ2
+ 2
⟨
(EL −E)Ψ′
Ψ
⟩
Ψ2
= 〈OZVZBL 〉Ψ2
(6)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 8 / 41
Zero-Variance, Zero-Bias Estimators
For observables which do not commute with H,
δ〈OL〉 = O(|δΨ|)
σ2(OL) = O(1)(7)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 9 / 41
Zero-Variance, Zero-Bias Estimators
For observables which do not commute with H,
δ〈OL〉 = O(|δΨ|)
σ2(OL) = O(1)(7)
But for the new estimator,
δ〈OZVZBL 〉 = O(|δΨ|2 + |δΨ||δΨ′|)
σ2(OZVZBL ) = O(|δΨ|2 + |δΨ′|2 + |δΨ||δΨ′|)
(8)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 9 / 41
ZVZB Recipe
Modify local estimator
OZVZBL = OL +∆OZV
L +∆OZBL (9)
Define Q = Ψ′/Ψ.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 10 / 41
ZVZB Recipe
Modify local estimator
OZVZBL = OL +∆OZV
L +∆OZBL (9)
Define Q = Ψ′/Ψ.
Zero-Variance term:
∆OZVL = −
1
2
(
∑k
∇2kQ + 2vi ·∇k Q
)
(10)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 10 / 41
ZVZB Recipe
Modify local estimator
OZVZBL = OL +∆OZV
L +∆OZBL (9)
Define Q = Ψ′/Ψ.
Zero-Variance term:
∆OZVL = −
1
2
(
∑k
∇2kQ + 2vi ·∇k Q
)
(10)
Zero-Bias term:∆OZB
L = 2(EL −〈EL〉)Q (11)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 10 / 41
Toulouse et al. approach
Choose Q such that ∇2Q term in ZV correction cancels delta-function inoriginal estimator.
Use generalised identity:
g(rij ,u)∇2ij
1
|rij −u|= −4πδ(rij −u) (12)
where g(rij ,u = rij) = 1.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 11 / 41
Toulouse et al. approach
Choose Q such that ∇2Q term in ZV correction cancels delta-function inoriginal estimator.
Use generalised identity:
g(rij ,u)∇2ij
1
|rij −u|= −4πδ(rij −u) (12)
where g(rij ,u = rij) = 1.
Suitable choice for spherically averaged intracule is
Q = −1
8π ∑i 6=j
Z
g(rij ,u)
|rij −u|
dΩu
4π(13)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 11 / 41
Toulouse et al. approach
Choose Q such that ∇2Q term in ZV correction cancels delta-function inoriginal estimator.
Use generalised identity:
g(rij ,u)∇2ij
1
|rij −u|= −4πδ(rij −u) (12)
where g(rij ,u = rij) = 1.
Suitable choice for spherically averaged intracule is
Q = −1
8π ∑i 6=j
Z
g(rij ,u)
|rij −u|
dΩu
4π(13)
g chosen to give correct asymptotic behaviour
g(rij ,u) = e−α|rij−u| (14)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 11 / 41
Toulouse et al. approach
Leads to the expressions
IZV (u) = −1
4π ∑i 6=j
[
vi ·rij
r2ij
(
A(rij ,u)+1
uB(rij ,u)
)
−α2
2A(rij ,u)
]
(15)
∆IZB(u) = −1
4π(EL −E)∑
i 6=j
A(rij ,u) (16)
where A,B are simple functions of exponentials.
Low variance, smooth, and can sample regions where electrons never go.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 12 / 41
A Note on Zero-Bias
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 0.5 1 1.5 2
I(u)
u (a.u.)
AccurateHartree-Fock
Hartree-Fock+ZB
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 13 / 41
A Note on Zero-Bias
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 0.2 0.4 0.6 0.8 1
I(u)
u (a.u.)
VMC poorVMC poor +ZB
AccurateDMC poor ZVZB
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 14 / 41
Spherical Coulomb hole in He2
-0.15
-0.1
-0.05
0
0.05
0.1
12 10 8 6 5.6 4 2
Sph
eric
al C
oulo
mb
hole
(a.
u.)
u (a.u.)
DMC ZVZB
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 15 / 41
Spherical Coulomb hole in He2
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
3 4 5 6 7 8 9 10 11 12
Sph
eric
al C
oulo
mb
hole
(a.
u.)
u (a.u.)
VMC ZVVMC ZVZBDMC ZVZB
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 16 / 41
Anisotropic Intracule and Extracule Densities
As a tool for understanding molecular electronic structure, sphericalaverage washes out lots of information.
Anisotropic intracule and extracule densities tell us about electron-pairorientation.
Can we simply use the approach of Toulouse et al. without performingthe spherical average?
Choose Q so that ∇2k Q term cancels delta-function.
Suitable choice is
Q = −1
8π ∑i 6=j
g(rij ,u)
|rij −u|(17)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 17 / 41
Intracule Density
Plugging Q into ZV expression gives
IZV (u) =1
8π ∑i 6=j
[
α2
|rij −u|e−α|rij−u| +
e−α|rij−u|
|rij −u|2vi · (rij −u)
(
1
|rij −u|−α)
+e−α|rji−u|
|rji −u|2vi · (rji −u)
(
1
|rji −u|−α)]
This ZV estimator has unpredictable error bars.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 18 / 41
Behaviour of First ZV Estimator
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2 2.5 3 3.5 4
4pi u
^2 I(
u)
u (a.u.)
ZV1a
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 19 / 41
Intracule Densities
Averaging this expression to get the spherical intracule removes nastybehaviour.
In 3D, we can obtain a well-behaved estimator by considering thecomplete expectation value of IZV ,
Z
Ψ2T IZV (u)dr1dr2 . . .drN (18)
and performing an integration by parts on the troublesome terms.Requires representation in terms of single-particle derivatives.
Equivalent estimator is
IZV (u) = −1
8π ∑i 6=j
(
∇2i Ψ
Ψ+ vi ·vi
)
[
e−α|rij−u|
|rij −u|+
e−α|rji−u|
|rji −u|
]
−α2
|rij −u|e−α|rij−u|
(19)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 20 / 41
Behaviour of Both ZV Intracule Estimators
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2 2.5 3 3.5 4
4pi u
^2 I(
u)
u (a.u.)
ZV1aZV2b
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 21 / 41
Linear Molecules
Cylindrical coordinates (ρ,φ,z). Align molecule along z-axis.
Cylindrical symmetry, so I(u) and E(R) reduce to 2D quantities.
To retain probabilistic interpretation,
F(ρ,z) = 2πρI(ρ,z) (20)
Prob. of finding electron pair a distance z apart along the bond, andsimultaneously a distance ρ apart perpendicular to the bond.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 22 / 41
Interpretation: RHF H2, R = 12.0 a.u.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 23 / 41
Application to Stretched H2 moleculeRassolov et al. 3 noticed interesting behaviour in the correlation energy ofH2 centered at around r = 2.5 a.u.
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0 1 2 3 4 5 6
Cor
rela
tion
ener
gy (
Ha)
r (a.u.)
3V.A.Rassolov, M.A.Ratner, J.A.Pople, J. Chem. Phys 112 4014 (2000)Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 24 / 41
Hartree-Fock Basis and Trial Wavefunction
Optimized Gaussian basis of 20s11p6d . EHF within 1×10−7 Hartree ofHF limit.
Usual DTN Jastrow 4 not as good as expected for H2. At equilibriumgeometry gives -1.17353(1) Ha compared to accurate value of -1.174475Ha.
DTN Jastrow includes EE, EN, EEN correlation.
4N.D.Drummond, M.D.Towler, R.J.Needs, PRB 70 235119 (2004)Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 25 / 41
Hartree-Fock Basis and Trial Wavefunction
Wavefunction of Alexander+Coldwell 5 can be written in Slater-Jastrowform. Slater part is product of 1s Slaters orbitals.
This ENEN Jastrow fully correlates electron-pairs and nuclei in scaledcoordinates:
JABij = ∑
lmnpq
γABlmnpq r l
iA rmjA rn
iB rpjB rq
ij (21)
where scaled coordinates are
r = r/(1+ αr) (22)
and l + m+ n + p + q ≤ N.
5S.A.Alexander and R.L.Coldwell, J. Chem. Phys 121 11557 (2004)Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 26 / 41
Wavefunction Quality
Table: VMC Energies (in Ha) of H2
Bond Length (a.u.) DTN(14;14;4,4) AC(N=6) KW 6
1.4 -1.17353(1) -1.1744756(7) -1.17447442.0 -1.13374(2) -1.1381318(8) -1.13813122.5 -1.06481(3) -1.093936(2) -1.09392733.0 -1.02422(3) -1.057317(2) -1.0573118
Quality of DTN Jastrow decreases as nuclei pulled apart. ENEN correlationimportant here.
6W.Kolos and L.Wolniewicz, J. Chem. Phys 43 2429 (1965)Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 27 / 41
Possibly interesting quantities
Average interparticle distance, 〈rij 〉.
Spherically averaged intra/extracule densities I(u),E(R), and correlationshifts.
Full 2D intra/extracule densities and correlation shifts.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 28 / 41
Spherically-averaged Intracule Density, R = 1.4 a.u.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8 9
4 pi
r^
2 I(
u)
u (a.u.)
~ExactUHF
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 29 / 41
Spherically-averaged Intracule Density, R = 2.0 a.u.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8 9
4 pi
r^
2 I(
u)
u (a.u.)
~ExactUHF
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 30 / 41
Spherically-averaged Intracule Density, R = 2.5 a.u.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8 9
4 pi
r^
2 I(
u)
u (a.u.)
~ExactUHF
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 31 / 41
Spherically-averaged Intracule Density, R = 3.0 a.u.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8 9
4 pi
r^
2 I(
u)
u (a.u.)
~ExactUHF
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 32 / 41
Anisotropic Intracule Density, R=1.4 a.u.
Figure: Contours at 0.115,0.08,0.02
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 33 / 41
Anisotropic Intracule Density, R=2.0 a.u.
Figure: Contours at 0.075,0.05,0.02
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 34 / 41
Anisotropic Intracule Density, R=2.5 a.u.
Figure: Contours at 0.062,0.04,0.02
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 35 / 41
Anisotropic Intracule Density, R=3.0 a.u.
Figure: Contours at 0.06,0.04,0.02
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 36 / 41
Interpretation
Anisotropic intracule density clearly shows the strong effect of correlationin the region 2.0−3.0.
Uncorrelated electrons lead to a delocalisation of the intracule densityaround R = 2.5, but with the peak at z = 0.
Correlation reduces the chance of finding electrons at small relative z.
UHF too ionic in this region, doesn’t want to dissociate cleanly.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 37 / 41
Bonding Properties
Extension to consider bonding properties is also possible. Recently donefor spherically-averaged properties of atoms in molecules by Ugalde.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 38 / 41
Bonding Properties
Extension to consider bonding properties is also possible. Recently donefor spherically-averaged properties of atoms in molecules by Ugalde.
In QMC we can consider the binding of arbitrary fragments.
∆I(r) = Iint(r)− Inon−int(r) (23)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 38 / 41
Bonding Properties
Extension to consider bonding properties is also possible. Recently donefor spherically-averaged properties of atoms in molecules by Ugalde.
In QMC we can consider the binding of arbitrary fragments.
∆I(r) = Iint(r)− Inon−int(r) (23)
One-body densities are easy. Consider two fragments, A and B.
nnon−int (r) = nA(r)+ nB(r) (24)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 38 / 41
Bonding Properties
Pair densities are more complicated, can’t simply add values fromisolated fragments.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 39 / 41
Bonding Properties
Pair densities are more complicated, can’t simply add values fromisolated fragments.
Wavefunction for non-interacting fragments is ΨAΨB . We need theintracule density according to this distribution.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 39 / 41
Bonding Properties
Pair densities are more complicated, can’t simply add values fromisolated fragments.
Wavefunction for non-interacting fragments is ΨAΨB . We need theintracule density according to this distribution.
As well as the contribution from intracules from systems A and B, we willneed cross terms:
1
2 ∑i∈A
∑j∈B
〈ΨAΨB|δ(rij −u)|ΨAΨB〉
〈Ψ2AΨ2
B〉(25)
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 39 / 41
Bonding Properties
Pair densities are more complicated, can’t simply add values fromisolated fragments.
Wavefunction for non-interacting fragments is ΨAΨB . We need theintracule density according to this distribution.
As well as the contribution from intracules from systems A and B, we willneed cross terms:
1
2 ∑i∈A
∑j∈B
〈ΨAΨB|δ(rij −u)|ΨAΨB〉
〈Ψ2AΨ2
B〉(25)
A full OO QMC code would be nice!
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 39 / 41
Summary
ZB correction not that useful for realistic simple trial wavefunctions.
Preliminary work on He2 shows different behaviour of I(u) than FCI resultaround u = 5.6. Real or ΨT artefact?
Derived ZVZB expressions for the full 3D Intracule and Extraculedensities.
Application to stretched H2 helps explain increased importance ofcorrelation as molecule dissociates.
Possibility of using QMC to discover more about detailed interactionsbetween fragments.
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 40 / 41
Acknowledgements
Prof. Ian Snook, A.Prof Salvy Russo.
Australian Research Council
Manolo Per (RMIT University, Melbourne) Anisotropic electron-pair densities in QMC August 1, 2008 41 / 41
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