Chemical bond and entanglement of electrons in the hydrogen molecule
aa r X i v : . [ qu a n t - ph ] A ug Chemical bond and entanglement of electrons in the hydrogen molecule
Nikos Iliopoulos and Andreas F. Terzis ∗ Department of Physics, School of Natural Sciences, University of Patras, Patras 265 04, Greece (Dated: September 24, 2018)We theoretically investigate the quantum correlations (in terms of concurrence of indistinguishableelectrons) in a prototype molecular system (hydrogen molecule). With the assistance of the stan-dard approximations of the linear combination of atomic orbitals and the configuration interactionmethods we describe the electronic wavefunction of the ground state of the H molecule. Moreover,we managed to find a rather simple analytic expression for the concurrence (the most used measureof quantum entanglement) of the two electrons when the molecule is in its lowest energy. We havefound that concurrence does not really show any relation to the construction of the chemical bond. PACS numbers: 03.67.-a,05.30.-d,75.10.Pq
I. INTRODUCTION
During the last two decades a lot of work has beendone in the relatively new scientific field of quantum in-formation and quantum computation [1]. In this newfield of physics it became soon apparent that one of themost, if not the most, important physical quantity -ameasure of quantum correlations- is the quantum en-tanglement [1, 2]. Historically, entanglement is the firsttype among quantum correlations which studied inten-sively demonstrating the difference between classical andquantum mechanics. Dated back to the Schr¨odinger era,introduced by him to describe the correlations betweentwo particles that interact and then separate, as in theEPR expertiment. Actually, nowadays exist many mea-sures of quantum entanglement. Entanglement of for-mation (EoF) or its computationally equivalent, concur-rence (Con), is the most widespread among them [3]. Itis widely accepted that EoF is a very reasonable anddecent measure of quantum entanglement having the ad-vantage that, in most cases, it can be computed easily.However, it became soon apparent that there are otherquantum correlations which cannot be included in entan-glement. Quantum discord [4, 5] is the most well-studiedmeasure for the total quantum correlations. Quantumdiscord takes into account all possible sources of quantumcorrelations but its computation, requiring optimaztionprocedures, is much more complicated and time consum-ing compared to the corresponding computation of theCon.All previously mentioned measures are well-studiedfor distinguishable particles. However, they are not sowidely used in indistinguishable particles. Especially forfermions (indistinguishable particles, having half-integerspin) obeying the Pauli exclusion principle, there aremany difficulties in order to implement the measures usedfor distinguishable particles. Schliemann et al. [6, 7]have, recently, presented a corresponding measure of Conof fermionin pure states, taking into account the con- ∗ [email protected] straints due to the Pauli exclusion principle.In the present study we investigate the simplest (pro-totype) realistic two- electrons molecular system, the hy-drogen molecule which has been extensively studied [8]for almost a century. Due to the well known principleof minimum energy the chemical bond is traditionallydescribed and specified in energy terms. Hence, we con-centrate in the study of the ground state of the molecularsystem which is the most stable state, especially in thelow temperature case where the thermal energy is notenough to excite the molecule. More specifically, in thepresent article, we investigate the possibility of any rela-tionship between quantum correlations and the chemicalbond.This paper is organized as follows: In section II wepresent the basic theory of indistinguishable particlesand we highlight the differences between indistinguish-able and distinguishable ones. In the same section, wefind an analytical expression for the Con of any two elec-trons being in a pure quantum state. In section III weapply this formula to the hydrogen molecule being in itsground state. Also, we apply the configuration interac-tion (CI) method in order to properly describe the groundstate of the hydrogen molecule. In the same section wepresent and analyze our results. Finally, we summarizeand conclude in section IV. II. THEORY
To begin with, we report some basic properties of in-dinguishable particles, which are necessary in order todevelop the quantum correlation identifiers. A generalpure state of two fermions is given by the equation [6, 7] | w i = n X i,j =1 w ij f † i f † j | i , (1) f † i , f † j are the single particle creation operators for twocorresponding subsystems acting on the vacuum stateand n is the dimensionality of the single-particle (one-electron) Hilbert space. Also, w is an antisymmetric co-efficient complex square matrix ( n × n ), with w ij = − w ji .The antisymmetric matrix fullfils the normalization con-dition T r ( w ∗ w ) = − / f i = P j U ij f ′ j , wetake new fermionic operators as well as new coefficientmatrix w ′ . So, the pure fermionic state of Eq. (1) takesthe form | w i = n X i,j =1 w ′ ij f ′ i † f ′ j † | i (3)where w ′ ij = ( U † wU ∗ ) ij . This new matrix have a blockdiagonal form [9], containing 2 × (cid:20) z k − z k (cid:21) (4)Each block has eigevalue z k . As a result the state of Eq.(1) takes the diagonal form | w i = 2 ≤ n X k =1 z k f ′† k − f ′† k | i (5)This is the equivallent of Schmidt decomposition in in-distinguishable particles [2]. Instead, now we have a sumof 2 × C ) for such apure state of eq. (1) with n = 4 is given by the equation C ( | w i ) = 8 | w w + w w + w w | , (6) w ij are elements of the antisymmetric matrix w . As inthe case of distinguishable particles, Con takes valuesfrom zero to one. If Con is zero then the fermionic statehas no entanglement and it has fermionic slater rank oneand could be represented by only one Slater determinant.Now, in order to find an equation for the entropy ofeach subsystem we have to trace out the other. Due tothe fact that fermions are indistinguishable particles thetwo parties have the same entropy. Also, the two fermiondensity matrix is given by ρ F = | w ih w | . According to R.Paskauskas and L. You [10] the single particle densitymatrix is ρ fνµ = T r ( ρ F f † µ f ν ) T r ( ρ F P µ f † µ f µ ) = 2( ω † ω ) µν (7) Now, the normalization condition takes the form ≤ n P k =1 | z k | = 1 /
4. Finally, the von Neumann entropy isgiven by the equation S f = − T r [ ρ f log( ρ f )] = − − ≤ n X k =1 | z k | log( | z k | ) (8)This single particle entropy ranges from unit to log( n E )and n E is the largest even number not larger than n.Both Con and entropy are measures of the entangle-ment of our system. However, concurrence ranges fromzero to unit, as in the case of separable particles, butentropy takes values 1 ≤ S f ≤ √ ( | i A ⊗ | i B + | i A ⊗ | i B ), and wewant to find the state of particle A then the von Neu-mann entropy is unit as the uncertainty about its stateis maximum. In the case of indistinguishable particles,we cannot separate them, thus there is extra uncertaintybecause we do not know to which particle we perform themeasurement. Nevertheless, this uncertainty does not af-fect the range of concurrence.From another point of view, the entanglement in thecase of sepable particles is due to spatial coordinates orspin. For instance, the states | i and | i above couldrepresent the possible states of spin (e.g. | i = | ↓i and | i = | ↑i ) or any other spatial separation. However,in order to have Con or entropy different than zero andone respectively, in the case of indistinguishable particles,the two particles should be entangled for spin and spatialcoordinates. A fine example is given by Eckert et al. [7].In the hydrogen molecule there are two nuclei (protons)as well as two electrons occupying the region outside theprotons. As a result each electron could be in nucleus Aor B and has spin | ↑i or | ↓i . Consequently, in our casethe single-particle (one- electron) Hilbert space is four-dimensional and this resulting in a six-dimensional two-particle (two- electrons) Hilbert space. For this reason,from now on, we will deal with the pure state of eq. (1)for n = 4. III. ENTANGLEMENT IN HYDROGENMOLECULE
Now, we turn our attention to the hydrogen molecule.As we said above, we have two electrons which wanderaround the two nucleus, A and B. Each electron couldhave spin +1/2 ( | ↑i ) or -1/2 ( | ↓i ). As a result thesingle-particle Hilbert space is four dimensional as it wasreported above ( | A i| ↑i = | i , | A i| ↓i = | i , | B i| ↑i = | i , | B i| ↓i = | i ).The Hamiltonian of our system is given by the equation(see for example, the classic book by P. Atkins and R.Friedman [8]) H = H + H + e πǫ R + e πǫ r (9)and H i = − ~ m e ∇ i − e πǫ r iA − e πǫ r iB , with i = 1 , r thedistance between the electrons. In addition, the two fi-nal terms represent the repulsive interaction between thenuclei and electrons respectively.As electrons are fermions, their wavefunction must beantisymmetric in order to obey in the Pauli exclusionprinciple. With the assistance of the linear combinationof atomic orbitals (LCAO) method [8] we conclude thatthe only possible wavefunctions of electrons areΨ = ψ + ψ + σ − (1 ,
2) (10a)Ψ = ψ − ψ − σ − (1 ,
2) (10b) ψ + = c + ( φ A + φ B ), ψ − = c − ( φ A − φ B ) and σ (1 , = √ ( | ↑i | ↓i −| ↓i | ↑i ). Here, φ A and φ B are the atomicorbitals of the hydrogen atom on ground state and c ± arenormaliztion factors which are c ± = √ ± S ) , where S isthe overlap integral (see the appendix). - E H H a r t r ee L FIG. 1. Plots of the average energy of the H molecule (actu-ally the energy difference of the H molecule minus the energyof two separated H atom) in Hartree units (i.e. devided bythe constant hcR H ) as a function of the internuclear distance( s = R/α , where α is the Bohr radius). The solid curvecorresponds to the energy (eq.(11)) of the molecule which iscomputed for a pure state described by eq.(10a). The dashed-dotted curve is the energy estimated for pure state describedby the linear combination (CI method) of eq.(12) Using the wavefunction, Ψ , of equation (10a) the hy-drogen molecule has an average energy given by the ex-pression E = 2 E s + j R − j ′ + 2 k ′ S + j + 2 k + m + 4 l S ) (11) where E s is the energy of the H atom in the ground stateIts plot as a function of the distance between the two nu-cleus is shows in Fig.(1) (solid curve). Then, if we find theaverage energy of the wavefunction given by the equation(10b), we find that these is not an absolute minimum, andthe energy is a decreasing function of the distance be-tween the two protons and finally taking a plateau valueequal to the plateau value of the energy of wavefunctionof equation (10a). For all distances the average energy ofthe Ψ state is larger than the energy of the wavefunctionof equation (10a). However, we can achieve a lower aver-age energy for the hydrogen molecule by describing themolecule by a wavefunction which is a linear combinationof the two previously mentioned wavefunctions. This isthe well known configuration interaction (CI) method [8].Hence, in the CI method the wavefunctions is describedby the following expressionΨ = c Ψ + c Ψ (12)with | c | + | c | = 1.Using the wavefunction given by equation (12) we canfind lower values for the energy of the ground state of thehydrogen molecule. The energy of the H molecule is forall distances lower than the energy of the wavefunctionΨ as it is indicated in Fig. 1 (compare the dashed-dottedto the solid curve). Its minimum is at approximately − .
237 Hartree ( ≈ − .
457 eV) for R ≈ . α .At this paragraph we show how in the context of thelinear combination state (CI method) we estimate thecoefficients c and c in order to achieve the lowest energyfor any given internuclear distance R . The expectationvalue of energy is given by E = | c | H + c c ∗ H + c ∗ c H + | c | H , where H (= E ) is the energy givenby eq.(11) and H = m − j − S ) , (13a) H = m − j − S ) , (13b) H = 2 E s + j R − j ′ − k ′ S + j + 2 k + m − l S ) (13c)where we point out that the H ij are functions of thedistance s . Now as | c | + | c | = 1, we can set c =cos ω and c = sin ω , assuming only real coefficients. Theminimization procedure is taken by the vanishing of thederivative ∂E/∂ω = 0. Consequently, we take the valuesof c and c for which the energy is least in every givendistance R . More specifically, the coefficients c and c are given by the following equations c = 12 + 12 q H H − H ) (14a) c = 12 − q H H − H ) (14b)Figure 2 shows the dependence of the values of thesquares of the c i ’s as a function of the internuclear dis-tance. Note that the minimization predict negative val-ues for the c . c i FIG. 2. Plots of the squares of the coefficients c i ( c i ) for i =1 , s = R/α between the two protons in the H molecule. The upper (lower) curve describes the c ( c )coefficient. Now, we turn our attention to the quantum entan-glement that the two electrons may have in the groundstate. As it was mentioned above, the wavefunction oftwo fermions is given by the eq. (1). Also the single-particle space is four-dimensional and more specifically {| i , | i , | i , | i} . However, in our case we have to set theelements w and w of antisymmetric matrix equal tozero because in the ground state the total spin of elec-trons must be zero. As a result the antisymmetric coef-ficient matrix w has the form w w − w w − w w − w − w (15)Additionally, concurrence from eq. (6) has the form C ( | w i ) = 8 | w w + w w | (16)By writing again the wavefunction for the ground stateof the hydrogen molecule from eq. (12) we assign thecoeffients of each state with the corresponding ones frommatrix of eq. (15) and we find that w = w = c c + c c − , (17a) w = − w = c c + c c − C ( | w i ) = 2 | c c | (18) C on c u rr e n ce FIG. 3. Concurrence as a function of the coefficient c for theground state of the hydrogen molecule in the CI method. Figure 3 shows how Con varies for different values of c . It is obvious that Con takes its maximum value for c = √ . However, from Figs. 1 and 2 we see that en-ergy takes its minimum value for a different value of c .The c = √ corresponds to internuclear distances R > larger than 8 α (i.e rather large distances). This fact is,also, amply demonstrated in Figure 4 as we can see howthe energy and Con vary as a function of the internu-clear distance. It is evident that Con takes small valuesfor small internuclear distances and becomes bigger asthe distance R is rising. Finally, it becomes essentiallyunit for R > α .The internuclear distance for which themolecule has its minimum energy, i.e. has its most sta-ble form, is R ≈ . α . For this distance Con takes thevalue 0.2378. - E H H a r t r ee L . C on c u rr e n ce FIG. 4. This figure shows Con (dashed curve) and energy(solid curve) as a function of the internuclear distance. Theenergy have been calculated by CI method.
As a result, one may conclude that quantum entan-glement is uncorrelated concerning the construction of achemical bond.Furthermore, we know from [9] that the product ofelements w w represent the entanglement which arisesdue to the orbital degrees of freedom and the product w w the corresponding entanglement due to the spindegrees of freedom. It is easily seemed that in our caseboth kinds of entanglement contribute to concurrence.To be more specific the fact the Con does not take itsmaximum value at the most stable state of the hydrogenmolecule, most probably means that it is not the propermeasure of the quantum correlations. Actually, this iswhat is now believed by the scientific community. Quan-tum discord provides all the quantum correlations andnot simply the Con [11, 12]. This could be one of the fu-ture plans, as for the moment these is no straightforwarddefinition for the quantum discord for indistinguishableparticles. Another future project, could be to study thecase of finite temperature, where not only the groundstate is occupied. In this case we believe a more obviousmeasure of the quantum correlations will be the entropy(see section II), as we now have a statistical mixtures.Hence, the chemical bond will correspond to maximiza-tion of a physical quantity that depend on both the en-ergy and the entropy (i.e. the free energy of the system). IV. CONCLUSIONS
In conclusion, we have presented a review of the the-ory for indistinguishable particles and we have reportedtwo measures of quantum correlations, the concurrenceand the von Neumann entropy. Next, taking advantageof this theoretical knowledge we implemented it in theprototype two- electrons hydrogen molecule. With theassistance of two well known approximations in molcec-ular physics (LCAO and CI methods) we systematicallyinvestigated the ground state of the H molecule. As a re-sult, we managed to find a closed form expression whichestimates a typical measure of quantum entanglement(concurrence) for the two electrons when the moleculehas its lowest energy.Finally, we expect (mainly by physical intuition) thatthe ground state of the molecule should be an entangledone. However, we have shown that the concurrence (anextensively used measure of quantum entanglement), isunrelated to the construction of the chemical bond. Inthis paper we computed the entanglement of the electronswhich are in the hydrogen molecule as this is the simplesttwo- electrons system in the universe and the most wellstudied as well as we need the minimum number approx-imations in order to find its ground state wavefunctions.In addition, we believe that the present methodology andsystematic theoretical research can be also applied toother much more complicated molecular systems. APPENDIX
Here we give some extra information concerning theintegrals which we used in our computations. Again, wehave in all of the following equations that j = e πǫ aswell as s = R/α , with α the Bohr radius. Addition-ally, φ α and φ β are the atomic orbitals of each hydrogenatom in the ground state. The required integrals for allcomputations are: j ′ j = Z φ α (1) r b dτ = 1 R [1 − (1 + s ) e − s ] (19) k ′ j = Z φ α (1) φ β (1) r b dτ = 1 α (1 + s ) e − s (20) jj = Z φ α (1) φ β (2) r dτ dτ = 1 R − α ( 2 s + 114 + 32 s + 13 s ) e − s (21) kj = Z φ α (1) φ β (1) φ α (2) φ β (2) r dτ dτ = A ( s ) − B ( s )5 α (22) lj = Z φ α (1) φ α (2) φ β (2) r dτ dτ = 12 α [(2 s + 14 + 58 s ) e − s − ( 14 + 58 s ) e − s ](23) mj = Z φ α (1) φ α (2) r dτ dτ = 58 α (24)with A ( s ) = 6 s [( γ + ln s ) S − E (4 s ) S ′ + 2 E (2 s ) SS ′ ] (25) B ( s ) = [ −
258 + 234 s + 3 s + 13 s ] e − s (26) S ( s ) = (1 + s + 13 s ) e − s (27) S ′ ( s ) = S ( − s ) = (1 − s + 13 s ) e s (28)Finally γ is Euler’s constant and E ( x ) is the well-knownexponential integral which is given by the equation E ( x ) = Z ∞ x e − z z dz (29) [1] M.A. Nielsen and I.L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press, Cambridge, 2000). [2] J. Audretsch,
Entangled Systems: New Directions inQuantum Physics (Wiley-VCH, Weinheim, 2007).[3] W.K. Wootters, Phys. Rev. Lett. 80 2245 (1998).[4] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88 017901(2001).[5] L. Henderson and V. Vedral, J. Phys. A 34 6899 (2001).[6] J. Schliemann, J. Ignacio Cirac, M. Kus, M. Lewensteinand D. Loss, Phys. Rev. A 64 022303 (2001).[7] K. Eckert, J. Schliemann, D. Bruss and M. Lewenstein,Annals of Physics 299 88-127 (2002). [8] Peter Atkins and Ronald Friedman,