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

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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 quan- tum 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. Ó 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 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.07.045 ARTICLE IN PRESS 2 Z. Huang, S. Kais / Chemical Physics Letters 413 (2005) 1–5 0 1 seminal work of Einstein, Podolsky, and Rosen [16] mPÀ1 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 a<b;r<s method of measuring the electron correlation, we pres- ent the scheme to quantify entanglement based on the where |W0æ is the ground 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 = Àxb, a,andxi, i =0. 2 In the occupation number representation simpler approach based on the two-electron density ma- trix 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 ...iþÁÁÁ 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 mPÀ1 C q1 ¼ Tr1jUihUj B 2 C 0 1 B 04jx2;2iþ1j 00C mPÀ1 mPÀ1 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 @ mPÀ1 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- XmÀ1 XmÀ1 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 Þ¼ÀTrðq 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 Z. Huang, S. Kais / Chemical Physics Letters 413 (2005) 1–5 3 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 À1= 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 À E | and the von Neumann Entanglement aþ2B aÀ2B B 2a C B À 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.
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