Entanglement As Measure of Electron–Electron Correlation in Quantum Chemistry Calculations

ARTICLE IN PRESS

Chemical Physics Letters 413 (2005) 1–5 www.elsevier.com/locate/cplett

Entanglement as measure of electron–electron correlation in quantum chemistry calculations

Zhen Huang, Sabre Kais *

Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, IN 47907, United States

Received 19 March 2005; in ﬁnal form 12 July 2005

Abstract

In quantum chemistry calculations, the correlation energy is defined as the difference between the Hartree–Fock limit energy and the exact solution of the nonrelativistic Schro¨dinger equation. With this definition, the electron correlation effects are not directly observable. In this report, we show that the entanglement can be used as an alternative measure of the electron correlation in quantum chemistry calculations. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. As an example we calculate the entanglement for He atom and H2 molecule with different basis sets. 2005 Elsevier B.V. All rights reserved.

The Hartree–Fock self-consistent field approxima- The concept of electron correlation as defined in tion, which is based on the idea that we can approx- quantum chemistry calculations is useful but not directly imately describe an interacting fermion system in observable, i.e., there is no operator in quantum terms of an effective single-particle model, remains mechanics that its measurement gives the correlation en- the starting point and the major approach for quanti- ergy. In this Letter, we propose to use the entanglement tative electronic structure calculations. In quantum as a measure of the electron correlation. Entanglement is chemistry calculations, the correlation energy is directly observable and it is one of the most striking defined as the energy error of the Hartree–Fock wave properties of quantum mechanics. function, i.e., the difference between the Hartree–Fock It was nearly 70 years ago when Schro¨dinger gave limit energy and the exact solution of the nonrelativis- the name ÔentanglementÕ to a correlation of quantum tic Schro¨dinger equation [1]. There also exists other nature. He stated that for an entangled state Ôthe best measures of electron correlation in the literature such possible knowledge of the whole does not include the as the statistical correlation coefficients [2] and more best possible knowledge of its partsÕ [7]. Over the dec- recently the Shannon entropy as a measure of the cor- ades the meaning of the word ÔentanglementÕ has relation strength [3,4]. Electron correlations have a changed its flavor and our view of the nature of strong influence on many atomic, molecular [5], and entanglement may continue to be modified [8]. Entan- solid properties [6]. Recovering the correlation energy glement is a quantum mechanical property that de- for large systems remains one of the most challenging scribes a correlation between quantum mechanical problems in quantum chemistry. systems that has no classical analog [9–12]. A pure state of a pair of quantum systems is called entangled if it is unfactorizable, as for example, the singlet state 1ffiffi * of two spin-1/2 particles, p ðj"#i j#"iÞ. A mixed Corresponding author. Fax: +765 494 0239. 2 E-mail addresses: [email protected], [email protected]. state is entangled if it can not be represented as a edu (S. Kais). mixture of factorizable pure states [13–15]. Since the

2 Z. Huang, S. Kais / Chemical Physics Letters 413 (2005) 1–5 0 1 seminal work of Einstein, Podolsky, and Rosen [16] mP1 B 4 jx j2 0 C there has been a quest for generating entanglement B 2;2iþ1 C q ¼ B i¼1 C. ð5Þ between quantum particles [10,17]. Investigation of n1;1 @ Pm A 2 quantum entanglement is currently a very active area 04jx1;2ij and has been studied intensely due to its potential i¼2 applications in quantum communications and infor- The matrix elements of x can be calculated from the mation processing [10] such as quantum teleportation expansion coefficient of the ab initio Configure Interac- [18,19], superdense coding [20], quantum key distribu- tion (CI) method. The CI wave function with single and tion [21], telecoloning, and decoherence in quantum double excitation can be written as X X computers [22–24]. jUi¼c jW iþ cr jWr iþ cr;s jWr;s i; ð6Þ In order to employ the entanglement as an alternative 0 0 a a a;b a;b ar aground state Hartree–Fock wave func- r entanglement measure for two-particle systems [25,26]. tion, ca is the coefficient for single excitation from orbi- r;s We will obtain a general approach to quantify the entan- tal a to r,andca;b is the double excitation from orbital a glement between different spin-orbitals of atomic and and b to r and s. Now the matrix elements of x can be molecular systems. written in terms of the CI expansion coefficients For two electron system in 2m-dimensional spin- 2iþ1 c0 c y 1 space orbital with c and c denote the fermionic annihi- x1;2 ¼ ; x2;2iþ1 ¼ ; a a 2 2 lation and creation operators of single-particle states c2iþ2 c2iþ1;2jþ2 and |0 represents the vacuum state, a pure two-electron 2 1;2 æ x1;2iþ2 ¼ ; x2iþ1;2jþ2 ¼ ; ð7Þ state |Uæ can be written as 2 2 X where i, j =1,2,..., m. In this general approach, the y y jUi¼ xa;bcacbj0i; ð1Þ ground state entanglement is given by von Neumann a;b2f1;2;3;4;...;2mg entropy of the reduced density matrix qn1, where a, b run over the orthonormalized single particle Sðqn Þ¼Trðqn log2qn Þ. ð8Þ states, and Pauli exclusion requires that the 2m · 2m 1 1 1 expansion coefficient matrix x is antisymmetric: We are now ready to evaluate the entanglement for the H molecule as a function of R using a direct and xa, b = xb, a,andxi, i =0. 2 In the occupation number representation simpler approach based on the two-electron density matrix calculated from the CI wave function with single (n1›, n1fl, n2›, n2fl, ..., nm›, nmfl), where › and fl mean a and b electrons, respectively, the subscripts denote the and double electronic excitations. In the occupation spatial orbital index and m is the total spatial orbital number representation, the CI wave function is given by number. By tracing out all other spatial orbitals except 3 4 jUi¼c0j1100 ...iþc1j0110 ...iþc2j1001 ...i n1, we can get a (4 · 4) reduced density matrix for the þ c3;4j0011 ...iþ ð9Þ spatial orbital n1 1;2 q Tr U U By tracing out all other orbitals except 1, we can get the n1 ¼ n1 j ih j 0 1 reduced density matrix for (n1› =0,1) q 000 B n1;0 C B mP1 C q1 ¼ Tr1jUihUj B 2 C 0 1 B 04jx2;2iþ1j 00C mP1 mP1 2 2iþ1;2iþ2 2 ¼ B i¼1 C; ð2Þ B jc2iþ1j þ jc j 0 C B m C 1 1;2 B P C B i¼1 i¼1 C B 004jx j2 0 C ¼ B C. @ 1;2i A @ mP1 A i¼2 0 jc j2 þ jc2iþ2j2 q 0 2 00 0n1;2 i¼1 ð10Þ q Ôempty orbitalÕ where n1;0 denotes an , The CI wave function expansion coefficients are calcu- Xm1 Xm1 q x 2 lated with the electronic structure package GAUSSIAN n1;0 ¼ 4 j 2iþ1;2jþ2j ð3Þ i¼1 j¼1 [27]. Thus, the entanglement of H2 molecule is readily calculated by the von Neumann entropy and qn1,2 denotes Ôtwo electron occupied orbitalÕ, Sðq Þ¼Trðq log q Þ. ð11Þ 2 1 1 2 1 qn1;2 ¼ 4jx1;2j . ð4Þ Fig. 1 shows the calculated entanglement S for H2 The Ôone electron occupied orbitalÕ,in(›, fl) basis set is molecule, Eq. (11), as a function of the internuclear given by distance R using Gaussian basis set 3-21G [27]. For ARTICLE IN PRESS

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1 Our two spin problem admits an exact solution, it is 0.05 H2 - Molecule simply a (4 · 4) matrix with the following four 0.8 eigenvalues: 0.04 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 0.6 k1 ¼J; k2 ¼ J; k3 ¼ 4B þ J c ; 0.03 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lement (S) 2 2 2 Ec g k ¼ 4B þ J c ð13Þ 0.4 4 0.02 Entan

Correlation (Ec) S and the corresponding eigenvectors: 0.01 0.2 0 1 0 1 0 0 Req = 0.74 Å B pffiffiffi C B pffiffiffi C 0 0 B 1= 2 C B 1= 2 C 012345 B pffiffiffi C B pffiffiffi C j/1i¼@ A; j/2i¼@ A; R (Å) 1= 2 1= 2 0 0 Fig. 1. Comparison between the absolute value of the electron 0 qffiffiffiffiffiffiffiffi 1 0 qffiffiffiffiffiffiffiffi 1 Exact UHF correlation Ec =|E E | and the von Neumann Entanglement aþ2B a2B B 2a C B 2a C (S) as a function of the internuclear distance R for the H2 molecule B C B C using Gaussian basis set 3-21G. At the limit R = 0, the dot represents B 0 C B 0 C j/ i¼B C; j/ i¼B C; ð14Þ the electron correlation for the He atom, Ec = 0.0149 (a.u.) using 3 B C 4 B C 3-21G basis set compared with the entanglement for He atom @ qffiffiffiffiffiffiffiffi0 A @ qffiffiffiffiffiffiffiffi0 A S = 0.0313. The equilibrium distance using 3-21G basis set is a2B aþ2B R = 0.74 A˚ . 2a 2a eq pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ 4B2 þ J 2c2. In the basis set {|››æ,|›flæ, |fl›æ,|flflæ}, the eigenvectors can be written as comparison we included the usual electron correlation Exact UHF (Ec =|E E |) and spin-unrestricted Hartree– 1 j/ i¼pffiffiffi ðj#"iþ j"#iÞ; ð15Þ Fock (UHF) calculations [27] using the same basis set 1 2 in the figure. At the limit R = 0, the dot represents the 1 electron correlation for the He atom, E = 0.0149 j/ i¼pffiffiffi ðj#"i j"#iÞ; ð16Þ c 2 2 (a.u.) using 3-21G basis set compared with the entangle- rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ment for the He atom S = 0.0313. With a larger basis a 2B a þ 2B j/3i¼ j##i þ j""i; ð17Þ set, cc-pV5Z [28], we obtain numerically Ec = 0.0415 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a rffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a (a.u.) and S = 0.0675. Thus, qualitatively entanglement a þ 2B a 2B j/ i¼ j##i j""i. ð18Þ and absolute correlation have similar behavior. At the 4 2a 2a united atom limit, R ! 0, both have small values, then rise to a maximum value and finally vanishes at the sep- Now we confine our interest to the calculation of the arated atom limit, R !1. However, note that for entanglement between the two electronic spins. For sim- R >3A˚ the correlation between the two electrons is plicity we take c = 1, Eq. (12) reduces to the Ising model almost zero but the entanglement is maximal until with the ground state energy k3 and the corresponding around R 4A˚ , the entanglement vanishes for R >4A˚ . eigenvector j/3æ. All the information needed for quanti- To understand the entanglement behavior for H2 fying the entanglement in this case is contained in the molecule using ab initio quantum chemistry methods, two-electron density matrix. we calculate the entanglement for a simpler two-electron When a biparticle quantum system AB is in a pure model system. This is a model of two spin-1/2 electrons state there is essentially a unique measure of the with an exchange coupling constant J (a.u.) in an effec- entanglement between the subsystems A and B given tive transverse magnetic field of strength B (a.u.). In by the von Neumann entropy S [29]. If we denote order to describe the environment of the electrons in a qA the partial trace of qAB with respect to subsystem molecule, we simply introduce a small effective external B, qA = TrB(qAB), the entanglement of the state qAB is magnetic field B. The general Hamiltonian for such a defined as the von Neumann entropy of the reduced system is given by density operator qA, S(qAB) ” Tr[qA log2qA]. For our model system in the ground state j/3æ, the J x x J y y z reduced density matrix in the basis set (›,fl) is given H ¼ ð1 þ cÞr1 r2 ð1 cÞr1 r2 Br1 2 2 by z ! I2 BI1 r ; ð12Þ 2 aþ2h 0 q ¼ 2a . ð19Þ ra a x y z c A a2h where are the Pauli matrices ( = , , )and is the 0 2a degree of anisotropy. For c = 1 Eq. (12) reduces to the Ising model, whereas for c = 0 it is the XY model. Thus, the entanglement is simply given by ARTICLE IN PRESS

4 Z. Huang, S. Kais / Chemical Physics Letters 413 (2005) 1–5 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 see, in the limit R ! 0, the entanglement converges to 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 p4ffiffiffiffiffiffiffiffiffiffiffiffiffiþ k 2 S ¼ log2 þ log2 ; the He atom results in ab initio methods but disappears 2 4 4 þ k2 2 2 4 þ k 4 þ k þ 2 in this simple model. ð20Þ Recently, a new promising approach is emerging for where k = J/B. the realization of quantum chemistry calculations with- The value of J, the exchange coupling constant be- out wave functions through first order semidefinite tween the spins of the two electrons, can be calculated programming [31]. The electronic energies and proper- as half the energy difference between the lowest singlet ties of atoms and molecules are computable simply from and triplet states of the hydrogen molecule. Herring an effective two-electron reduced density matrix q(AB). Thus, the electron correlation can be directly calculated and Flicker have shown [30] that J for H2 molecule can be approximated as a function of the interatomic as effectively with the entanglement between the two distance R. In atomic units, the expression for large R electrons, which is readily calculated as the von Neu- is given by mann entropy S = TrqA log2qA, where qA = TrBq(AB). Utilizing this combined approach, one calculates the 5=2 2R 2 2R JðRÞ¼0.821R e þ OðR e Þ. ð21Þ electronic energies and properties of atoms and mole- Fig. 2 shows the calculated von Neumann Entangle- cules including correlation without wave functions or ment (S), Eq. (20), as a function of the distance between Hartree–Fock reference systems. the two electronic spins R, using J(R) of Eq. (21), for In summary, we presented the entanglement as an different values of the magnetic field strength B. At the alternative measure of the electron electron correlation limit R !1 the exchange interaction J vanishes as a in quantum chemistry calculations for atoms and mole- result the two electronic spins are up and the wave func- cules. All the information needed for quantifying the tion is factorazable, i.e., the entanglement is zero. At the entanglement is contained in the two-electron density other limit, when R = 0 the entanglement is zero for this matrix. This measure is readily calculated by evaluating model because J =0.AsR increases, the exchange inter- the von Neumann entropy of the one electron reduced action increases leading to increasing entanglement density operator. This definition of correlation has deep between the two electronic spins. However this increase roots in quantum theory, observable, and does not need in the entanglement reaches a maximum limit as shown a reference system such as Hartree–Fock calculations. in the figure. For large distance, the exchange interac- The approach is general and can be used for larger tion decreases exponentially with R and thus the atomic and molecular systems. decrease of the entanglement. The figure also shows that the entanglement increases with decreasing the magnetic Acknowledgments field strength. This can be attributed to effectively increasing the exchange interaction. Thus, we get the We thank Professor Dudley Herschbach for critical similar behavior as the entanglement for the H mole- 2 reading of the manuscript. This work has been cule as a function of the internuclear distance R, using supported by the Purdue Research Foundation. accurate ab initio methods. Because the Eq. (21) is only applicable to the large value of R, it is not surprising to References

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