Quantum chemistry and materials simulation in the age
of entanglement
Garnet Kin-Lic Chan Princeton University
The last time I did an experiment
!"#$%&'%#$(&)*$"+)$!,$*",-$.#$-"+!$!,$),/$
What we want to understand (I) biocatalysis
N2 + 8H+ + 8e + 16ATP !
2NH3 +H2 + 16ADP + 16Pi
0&!1,2#0$34+5,0$
H2O+ 4e + nh !
H2 +O26",!,*70!"#*&*$
",-$),#*$8+!91#$6#1:,1.$1#.+1(+;%#$
What we want to understand (II) molecular materials
",-$),$-#$.+(#$.+!#1&+%*$-&!"$.,%#
What we want to understand (III) high temperature superconductivity
?@ABC$
MNNAC$&1,0O;+*#)D$E
Quantum mechanics is the foundation of chemistry
P&1+
Quantum mechanics
H = E U$&*$+$)&V#1#05+%$,6#1+!,1$
*&02%#$6+15#0#12*$,:$!1+0*&5,0*$;#!-##0$)&V#1#0!$,1;&!+%*$ (r)
| (r)|2 61,;+;&%&!7$)&*!1&;95,0$,:$#%#
Q,1;&!+%R$
30#$*!19
Many particle QM
#$?*!$;+%%J*$6,*&5,0$!"#$%$"#$"&'(),:$M0)$;+%%J*$6,*&5,0$
,;*#1>#$ ,;*#1>#$ ,;*#1>#$ ,;*#1>#$
,;*#1>#)$>+%9#$,:$?*!$$6,*&5,0$#$%$"#*$,0$,;*#1>#)$
>+%9#$,:$M0)$6,*&5,0$
many particle wavefunctions
-+>#:90(r1, r2, ...rn)*&.9%!+0#,9*$61,;+;&%&!7$Q+.6%&!9)#R$,:$0$#%#
?P$ MP$ GP$ 0P$
KKK$
W9+0!9.$*&.9%+5,0$QL9+0!9.$#:90
Doing the best you can: Pople
,0#$#%#
^+*&*$Q*6+5+%$1#*,%95,0R$
U_$
E1#+502$.,1#$#%#
P,9;%#*$ E1&6%#*$ 8O!96%#*$
`,6%#$
,0#$#%#
E1#+502$.,1#$#%#
&0
but this view of many-particle QM is depressing
in general the many-electron wave func5on for a system of N electrons is not a legi5mate scien5fic concept [for large N]
Kohn (Nobel lecture, 1998)
[The Schrodinger equa5on] cannot be solved accurately when the number of par5cles exceeds about 10. No computer exis5ng, or that will ever exist, can break this barrier because it is a catastrophe of dimension ...
Pines and Laughlin (2000)
Density Functional Theory?
5,4"6*4/0)#$"*!&()71"83,"/')&4$,-()
+%*,$(0,-0$+*D$Yf&>&02$g6Z$
E[] = Emf
[] + Exc
[]
Y:,1$*7*!#.*$,:$>#17$.+07$#%##$:,1$#+&,91$&0)#6#0)#0!%7D$+))$Y
but there are many problems :90
Insights into Current Limitations ofDensity Functional TheoryAron J. Cohen, Paula Mori-Snchez, Weitao Yang*
Density functional theory of electronic structure is widely and successfully applied in simulationsthroughout engineering and sciences. However, for many predicted properties, there are spectacularfailures that can be traced to the delocalization error and static correlation error of commonly usedapproximations. These errors can be characterized and understood through the perspective of fractionalcharges and fractional spins introduced recently. Reducing these errors will open new frontiers forapplications of density functional theory.
Interactions between electrons determine thestructure and properties of matter from mol-ecules to solids. To describe interacting elec-trons, the extremely simple three-dimensionalelectron density can be used as the basic variablewithin density functional theory (DFT) (1, 2),negating the need in many cases for the massive-ly complex many-dimensional wave function.
Kohn noted in his Nobel lecture that DFThas been most useful for systems of very manyelectrons where wave function methods encounterand are stopped by the exponential wall (3). Thebeauty of DFT is that its formalism is exact yetefficient, with one determinant describing theelectron densityall of the complexity is hiddenin one term, the exchange-correlation functional.This term holds the key to the success or failureof DFT. Exchange arises from antisymmetry dueto the Pauli exclusion principle, and correlationaccounts for the remaining complicated many-body effects that need many determinants to befully described. However, the form of exchange-correlation in terms of the density remains un-known and it is necessary to use approximations,so that inmany casesDFTcontains semi-empiricalparameters. The great success of DFT is that sim-ple approximations perform remarkably well fora wide range of problems in chemistry and phys-ics (46), particularly for prediction of the struc-ture and thermodynamic properties of moleculesand solids.
Despite thewidespread popularity and successof DFT, its application can still suffer from largepervasive errors that cause qualitative failures inpredicted properties. These failures are not break-downs of the theory itself but are only due todeficiencies of the currently used approximateexchange-correlation functionals. A systematicapproach for constructing functionals that areuniversally applicable is a hard problem and hasremained elusive.
A possible path forward is to have a deeperlook at the errors embedded in currently usedfunctionals to determine the origin of their pa-thologies at the most basic level. Recent workhas traced many of the errors in calculations toviolations of conditions of the exact functional(7, 8). These violations present themselves inextremely simple model atoms, which can beused for diagnosis, and more importantly, inmany interesting and complex chemical and
physical systems. Identifying and understandingthe basic errors offer a much needed path for thedevelopment of functionals, as well as a usefulinsight into potential pitfalls for practicalapplications.
What are some of the major failures in DFTcalculations? First, they underestimate the barriersof chemical reactions, the band gaps of materials,the energies of dissociating molecular ions, andcharge transfer excitation energies. They also over-estimate the binding energies of charge transfercomplexes and the response to an electric field inmolecules andmaterials. Surprisingly, all of thesediverse issues share the same rootthe delocal-ization error of approximate functionals, due to thedominating Coulomb term that pushes electronsapart. This error can be understood from a per-spective that invokes fractional charges (7, 9).Furthermore, typical DFT calculations fail to de-scribe degenerate or near-degenerate states, suchas arise in transitionmetal systems, the breaking ofchemical bonds, and strongly correlated materials.All of these problems are merely manifestationsof another common errorthe static correlationerror of approximate functionals. This problemarises because of the difficulty in using the elec-tron density to describe the interaction of degen-
Department of Chemistry, Duke University, Durham, NC27708, USA.
*To whom correspondence should be addressed. E-mail:[email protected]
A H2+ binding curve
C H2 binding curve
B H atom with fractional charge
D H atom with fractional spins
R (Angstrom)
E
(kc
al/m
ol) 1 3 5 7 9
R (Angstrom)1 3 5 7 9
Fractional charge (e)
Delocalizationerror
Staticcorrelationerror
HFB3LYP
LDA
0.0 0.25 0.5 0.75 1.0
Fractional spins [,][1,0] [3/4,1/4] [1/2,1/2] [1/4,3/4] [0,1]
30
-30
-60
0
200
0
-100
100
Fig. 1. DFT approximations fail. The dissociation of H2+ molecule (A) and H2 molecule (C) are shown forcalculations with approximate functionals: Hartree-Fock (HF), local density approximation (LDA), andB3LYP. Although DFT gives good bonding structures, errors increase with the bond length. The huge errorsat dissociation of H2
+ exactly match the error of atoms with fractional charges (B), and for H2 they exactlymatch the error of atoms with fractional spins (D). The understanding of these failures leads to thecharacterization of the delocalization error and static correlation error that are pervasive throughoutchemistry and physics, explaining a host of problems with currently used exchange-correlation functionals.
8 AUGUST 2008 VOL 321 SCIENCE www.sciencemag.org792
Challenges in Theoretical Chemistry
),#*$0,!$
P_E$ P&T
illusion of complexity 0+!91#$),#*$0,!$#46%,1#$+%%$6,**&;&%&5#*$
#46,0#05+%$
physical many-electron wavefunctions
H(r1, . . . , rn) = E(r1, . . . , rn)
(r1, r2, ...rn)
*&.9%!+0#,9*$61,;+;&%&!7$Q+.6%&!9)#R$,:$0$#%#
21,90)$+0)$%,-O%7&02$L9+0!9.$*!+!#*$,:$U+.&%!,0&+0*$-&!"$%4(*!8/'$&0!#1+
Local realism recovered k#$#,$,;*#1>#$%,m:$-#$)&>&)#$*7*!#.$&0!,$&00#1$+0)$,9!#1$1#2&,0*D$,0%7$)#21##*$,:$:1##),.$$,0$!"#$;,90)+17$+1#$Y#0!+02%#)Z/$d91:+
General argument U+*502*C$Q:1,.$eOj,;&0*,0$;,90)*R$
exp(H)|iH)
|i 61,)9
;9!$U+.&%!,0&+0$+66%)$30&!#$$$ 5.#*$O(1/Gap)Gap)
,0%7$%&.&!#)$+.,90!$,:$#0!+02%#.#0!$2#0#1+!#)$
!19#$21,90)O*!+!#$
Mathematics of entanglement
0?$ 0M$&0)#6#0)#0!$
#0!+02%#)$ MM
(n1, n2) = A(n1)A(n2)
(n1, n2) =X
i
A(n1, i)A(n2, i)
%,-$#0!+02%#.#0!$Q6"7*&
Tensor networks
&0)#6#0)#0!$
(n1, n2, n3) = A(n1)A(n2)A(n3)
#0!+02%#)$ M(n1, n2, n3) =
X
ij
A(n1, i)A(n2, i, j)A(n3, j)
',:)$"&/"9'$0$"&
Tensor networks and Lego !#0*,1$0#!-,1(*C$6"7*:90
i,.69!+5,0$&0>,%>#*$
Simple example
r/re M=250 M=500 1.0 -0.41914 -0.41919 1.6 -0.65718 -0.65719 2.0 -0.91789 -0.91789 3.2 -2.67195 -2.67195
H H H H H H H H H H H H
H50 symmetric bond-stretch (50,50) ac5ve space 1028
e
