research article quantum-dynamical theory of electron ...electron spin and electron exchange...

Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2013, Article ID 497267, 8 pageshttp://dx.doi.org/10.1155/2013/497267

Research ArticleQuantum-Dynamical Theory of Electron Exchange Correlation

Burke Ritchie1,2 and Charles A. Weatherford3

1 Lawrence Livermore National Laboratory, Livermore, CA 94550, USA2 Livermore Software Technology Corporation, Livermore, CA 94550, USA3Department of Physics, Florida A&M University, Tallahassee, FL 32307, USA

Correspondence should be addressed to Burke Ritchie; [email protected]

Received 4 November 2012; Revised 13 January 2013; Accepted 14 January 2013

Academic Editor: Benjaram M. Reddy

Copyright © 2013 B. Ritchie and C. A. Weatherford. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The relationship between the spin of an individual electron and Fermi-Dirac statistics (FDS), which is obeyed by electrons in theaggregate, is elucidated.The relationship depends on the use of spin-dependent quantum trajectories (SDQT) to evaluate Coulomb’slaw between any two electrons as an instantaneous interaction in space and time rather than as a quantum-mean interaction inthe form of screening and exchange potentials. Hence FDS depends in an ab initio sense on the inference of SDQT from Dirac’sequation, which provides for relativistic Lorentz invariance and a permanent magnetic moment (or spin) in the electron’s equationof motion. Schroedinger’s time-dependent equation can be used to evaluate the SDQT in the nonrelativistic regime of electronvelocity. Remarkably FDS is a relativistic property of an ensemble of electron, even though it is of order 𝑐0 in the nonrelativistic limit,in agreement with experimental observation. Finally it is shown that covalent versus separated-atoms limits can be characterizedby the SDQT. As an example of the use of SDQT in a canonical structure problem, the energies of the 1Σ

𝑔and 3Σ

𝑢states of H

2are

calculated and compared with the accurate variational energies of Kolos and Wolniewitz.

1. Introduction

One may consider that quantum chemistry is dominated bytheoretical and computational efforts to achieve an accuratedescription of electron exchange correlation, evolving suchworkhorse methodologies as Hartree-Fock-ConfigurationInteraction, Density Functional Theory, and numerous vari-ations on the theme of nonrelativistic quantum mechanicsapplied to problems of chemical interest. But yet, owingto historical happenstance, more heat than light has beengenerated concerning the fundamental physical understand-ing of exchange-correlation. Even in early calculations inwhich correlation was built into the wave function it wasrecognized that the concept of exchange tended to losemeaning in a calculation in which correlation was treated tohigh accuracy [1]. As another example it was shown that ahigh-order perturbation calculation in which the electron-electron interaction is treated as the perturbation is able toachieve order by order the correct permutation symmetryof the wave function starting in zeroth order with a simpleunsymmetrized product of orbitals [2]. This conundrum

can be easily understood for the two-electron, one-nucleusproblem by a simple change from nucleus centered to Jacobicoordinates in which one vector connects the two electrons,and the other connects the center of mass of the two electronsto the nucleus (where of course it is understood that theBorn-Oppenheimer picture is being used). Then the wavefunction separates naturally into two linearly-independentwave functions which are either even or odd in the exchangeof electrons. In other words the new choice of coordinates inwhich correlation is achieved through a physically judiciouschoice of coordinates also achieves the correct exchangesymmetry, which is guaranteed by the symmetry of theHamiltonian with respect to the electron exchange.

Nonrelativistic quantum theory fails us, however, evenfor two-electron problems. According to experimental obser-vation, the electrons have intrinsic angular momentumcomprising two-spin-1/2 states, such that the total wavefunctionmust be antisymmetric on electron exchange, whichis satisfied either by a product of the even spatial wavefunction adduced above with a spin state which is odd onelectron exchange (singlet state) or by a product of the

2 Advances in Physical Chemistry

adduced odd spatial wave function with a spin state whichis even on electron exchange (triplet state). The general-ization is of course the Slater determinantal wave functionof spin orbitals which guarantees the correct antisymmetricpermutation symmetry for N spin-1/2 particles. Hence thestandard methodologies [3], for all their success, simulatemany-electron quantum states in an ad hoc manner, basedon experimental observation, but tell us nothing fundamentalabout the physical basis of how the spin state of an individualelectron is related to the Pauli exclusion principle and theobserved Fermi-Dirac statistical behavior of an aggregate ofelectrons.

2. Spin-Dependent Quantum Trajectories andElectron Exchange Correlation

Readers should recognize that an electron’s spin state and itsspatial correlation with another electron are strongly related,as suggested by our observation that a successful simulationof correlation is also accompanied by the correct exchangesymmetry. Notice, however, that such a relationship betweenelectron spin and the electron-electron Coulomb interactionappears to be grossly at oddswith our intuitive understandingof quantum chemistry, likely due to the absence of a particle-trajectory picture in the standard methodologies. Bohm for-mulated a quantum dynamical approach in which the phase-amplitude solution of Schroedinger’s equation has a formalrelationship to classical hydrodynamics [4, 5]. Bohmiandynamics has recently undergone a renewed interest [6].It is a method of solving the time-dependent Schrodingerequation by assuming a form for the wavefunction Ψ(𝑥, 𝑡) =𝑅(𝑥, 𝑡)𝑒

𝑖𝑆(𝑥,𝑡) where R (such that R is greater than or equalto zero for all x) is the amplitude and S is the actionfunction. Then upon substitution into the time-dependentSchroedinger equation, two coupled equations (continuityequation and the quantum Hamilton-Jacobi equation) areobtained. The net result is to produce quantum trajectories.These equations have proven to be difficult to solve accurately,particularly for multi electrons [6]. In fact, it is difficult tosolve in three spatial dimensions for one electron. Note thatin the context of the Bohmian equations, the discretizationis done in terms of what are called pseudo particles. Thus,when it is said that say fifty particles are propagated, ittypically means fifty pseudo particles are used to describethe dynamics of one real particle. As far as we are aware, noone has computed the dynamics of two real particles usingBohmian dynamics, and in fact, it does not seem possible todo so with current algorithms and computer hardware.

We therefore seek a quantum trajectory approach forreal electrons in which the spin of an individual electron iscorrectly accounted for, which means that we must look toDirac’s rather than to Schroedinger’s equation. One of thedesiderata of Dirac’s program [7] for incorporating Einstein’sspecial theory of relativity into quantum mechanics was thatthe quantum analog of the classical equation of continuity,

𝜕𝜌

𝜕𝑡

+ ∇⃗ ⋅ �⃗�= 0, (1)

should be inferred from the equation of motion for therelativistic electron,

(𝑖ℎ

𝜕

𝜕𝑡

− 𝑉 − 𝑚𝑐2)𝜓 ( ⃗𝑟, 𝑡) + 𝑖ℎ𝑐�⃗� ⋅ ∇⃗𝜒 ( ⃗𝑟, 𝑡) = 0, (2a)

(𝑖ℎ

𝜕

𝜕𝑡

− 𝑉 + 𝑚𝑐2)𝜒 ( ⃗𝑟, 𝑡) + 𝑖ℎ𝑐�⃗� ⋅ ∇⃗𝜓 ( ⃗𝑟, 𝑡) = 0, (2b)

whereupon [7],

⃗𝑗 ( ⃗𝑟, 𝑡) = 𝑐 [𝜓+( ⃗𝑟, 𝑡) �⃗�𝜒 ( ⃗𝑟, 𝑡) + 𝜒

+( ⃗𝑟, 𝑡) �⃗�𝜓 ( ⃗𝑟, 𝑡)] , (3)

where �⃗� is Pauli’s vector, and the quantum density is

𝜌 ( ⃗𝑟, 𝑡) = 𝜓+( ⃗𝑟, 𝑡) 𝜓 ( ⃗𝑟, 𝑡) + 𝜒

+( ⃗𝑟, 𝑡) 𝜒 ( ⃗𝑟, 𝑡) , (4)

where 𝜓( ⃗𝑟, 𝑡) and 𝜒( ⃗𝑟, 𝑡) are the large and small Diracspinors, respectively, and the superscripts denote Hermi-tian conjugates. On eliminating (2b) in favor of (2a) anddropping all terms of order 𝑐−2, which is the nonrelativisticlimit, Schroedinger’s equation is recovered for 𝜓, which wasanother desideratum in Dirac’s program for a relativistic-electron theory. Now notice that a velocity field can beinferred from the current by writing

⃗𝑗 ( ⃗𝑟, 𝑡) = 𝑐 [𝜓+( ⃗𝑟, 𝑡) �⃗�𝜒 ( ⃗𝑟, 𝑡) + 𝜒

+( ⃗𝑟, 𝑡) �⃗�𝜓 ( ⃗𝑟, 𝑡)]

= �⃗� ( ⃗𝑟, 𝑡) 𝜌 ( ⃗𝑟, 𝑡) ,

(5)

and solving for �⃗�( ⃗𝑟, 𝑡), from which a trajectory, ⃗𝑠(𝑡), can becalculated from the time integration of the velocity field tofind a position field,

�⃗� ( ⃗𝑟, 𝑡) = ∫

𝑡

0

𝑑𝑡�⃗� ( ⃗𝑟, 𝑡

) , (6)

and finally by finding the quantum expectation value of theposition field,

⃗𝑠 (𝑡) = ∫𝑑 ⃗𝑟 [𝜓+( ⃗𝑟, 𝑡) �⃗� ( ⃗𝑟, 𝑡) 𝜓 ( ⃗𝑟, 𝑡)

+𝜒+( ⃗𝑟, 𝑡) �⃗� ( ⃗𝑟, 𝑡) 𝜒 ( ⃗𝑟, 𝑡)] .

(7)

One more step is needed to show the relationship ofelectron spin and electron exchange correlation, namely,the evaluation of the electron-electron Coulomb interactionusing quantum trajectories. Considering the case of twoelectrons, one of which is described by (1)–(7). The Coulombinteraction for an electron with position vector ⃗𝑟 with theother electron with position vector ⃗𝑟 can then be written asfollows:

𝑉𝑒𝑒 ( ⃗𝑟, 𝑡) =

𝑒2

⃗𝑟 − ⃗𝑠(𝑡)

. (8)

Similarly the electron at ⃗𝑟 is described by (1)–(7) in which ⃗𝑟is replaced by ⃗𝑟 and the interaction is given by

𝑉𝑒𝑒( ⃗𝑟, 𝑡) =

𝑒2

⃗𝑟− ⃗𝑠 (𝑡)

. (9)

Advances in Physical Chemistry 3

−20

−10

0

10

20

Sy (a

u)

0 50 100 150 200𝑡 (au)

Figure 1: Quantum trajectories in the y direction at 𝑅 = 1.4 aushowing the correlation of the two electrons in the 1Σ

𝑔state of H

2.

Solid: spin-up electron. Dotted: spin-down electron.The trajectoriesare calculated from (7) in the nonrelativistic limit using spectralwave functions.

Thus the electron-electron potential is evaluated as aninstantaneous interaction in space and time rather thanas a quantum-mean interaction in the form of Coulomband exchange integrals. This is a critical step. Normally theinteraction is written as 𝜐

𝑒𝑒 = 𝑒

2/| ⃗𝑟 − ⃗𝑟

|, which is a func-

tion in 6-space well known to be incompatible with Lorentz-invariance [8], which is a requirement for a correct rela-tivistic theory. The authors’ comment [8], following theirequation (38.4), “the Coulomb term [e-e interaction] in(38.3), however, is not even approximately Lorentz invariant,and relativistic corrections to the interaction between the twoelectrons are furnished by quantum electrodynamics,” refersto the Lorentz-invariant form of the e-e interaction given byits representation in QED in terms of the exchange of virtualphotons. Notice that in the present paper we are claiming thatthese relativistic corrections, at least for nonradiative interac-tions, are given by representing the e-e or e-e interactions inthe forms given by (8)-(9) and by calculating the trajectoriesusing (2a)–(7). This is so because the 4-space form of (8)-(9) is fully compatible with the Lorentz invariance of Dirac’sone-electron equation as given by (2a) and (2b), in which Vis explicitly time dependent and can be written as 𝑉( ⃗𝑟, 𝑡). Inother words the time-independent 6-space potential, 𝜐

𝑒𝑒 =

𝑒2/| ⃗𝑟 − ⃗𝑟

|, is represented in 3-space and the time by using

the position vector of one electron and the trajectory of theother. Readers will note that Dirac’s one-electron equation,as given for example in the literature for the hydrogen atom,is exact and therefore Lorentz invariant for the Coulombattraction to the nucleus, 𝑉(𝑟) = −𝑍𝑒2/𝑟. It is still exact andLorentz invariant for the general scalar potential 𝑉( ⃗𝑟, 𝑡). Thisis a true statement because in the electron’s 4-momentum,(𝛾𝑚𝑐, 𝛾𝑚�⃗�) = (𝛾𝑚𝑐, �⃗�) = (𝛾𝑚𝑐, −𝑖ℎ∇⃗), the scalar part can

be expressed as terms of (1/𝑐)[𝑖ℎ(𝜕/𝜕𝑡) − 𝑉( ⃗𝑟, 𝑡)] on usingthe operator form of the classical relativistic Hamiltonian𝑖ℎ(𝜕/𝜕𝑡) = 𝛾𝑚𝑐

2+ 𝑉( ⃗𝑟, 𝑡), where 𝛾 is the Lorentz factor

𝛾 = √1 + (𝑝2/𝑚2𝑐2). We can only guess that workers have

not used the present methodology to find a fully relativistic,Lorentz-invariant theory for many fermions due historicallyto the dominance of time-independent quantum theory inabsence of an explicitly time-dependent interaction suchas the vector potential, �⃗�( ⃗𝑟, 𝑡), in matter-radiation physics,in which case the 4-momentum is written (𝛾𝑚𝑐, −𝑖ℎ∇⃗ −(𝑒/𝑐)�⃗�( ⃗𝑟, 𝑡)). Equations (8)-(9) are functions of 3-space andthe time and are fully compatible with the Lorentz invarianceof Dirac’s one-electron Hamiltonian and with the Lorentzcovariance of his one-electron wave equation [7]. Notice thatwe do not venture onto the ice of two-body Dirac theory [9].Proceeding heuristically we believe that it is possible to derivea correct many-electron relativistic quantum theory simplyby replacingNewton’s equation ofmotion for each electron inan aggregate of electrons by Dirac’s equation of each electronin the aggregate, thusly for an electron whose position vectoris ⃗𝑟 interacting with another electron whose position vectoris ⃗𝑟,

𝑑�⃗� (𝑡)

𝑑𝑡

= −∇⃗

𝑒2

⃗𝑟 (𝑡) − ⃗𝑟

(𝑡)

→

𝑖ℎ

𝜕𝜓𝐷 (

⃗𝑟, 𝑡)

𝜕𝑡

= [ − 𝑖ℎ𝑐�⃗� ⋅ ∇⃗ + 𝛽𝑚𝑐2

+

𝑒2

⃗𝑟 − ⃗𝑠(𝑡)

] 𝜓𝐷( ⃗𝑟, 𝑡) ,

(10)

where �⃗� = 𝛾𝑚(𝑑 ⃗𝑟(𝑡)/𝑑𝑡), 𝛾 = √1 + (𝑝2/𝑚2𝑐2), 𝜓𝐷

= (𝜓

𝜒 ),�⃗� = (

0 �⃗�

�⃗� 0), 𝛽 = ( 𝐼 0

0 −𝐼), and I is the 2 × 2 identity matrix.

Equations (2a) and (2b) are identical to the right-hand side ofthe arrow in (10), which is just another way of writing Dirac’sequation.

Pauli’s exclusion principle is obeyed if (2a) and (2b) andtheir counterparts for a second electron are evolved in thetime for identical spatial functions and opposite spin states atinitial time (Figure 1), which corresponds to a singlet stateor for different spatial functions and the same spin states atinitial time (Figure 2), which corresponds to a triplet state.Notice that Figure 1 shows quantum trajectories with slightlydifferent frequencies, which can happen because spatial wavefunctions which are identical at initial time can developdifferences over time, in analogy to the use of unrestrictedHartree-Fock theory in standard time-independent theory.It should be noted that, although one must choose a set ofinitial conditions for the time integration of the equations,the problem is not an initial-value problem in the same senseas in classical mechanics in which, due to the deterministicnature of the trajectories, one must find the physical resultas an average over a set of different initial conditions sincewe cannot know a unique set of initial conditions forthe particles. The reason can be found in the Heisenberguncertainty principle, Δ𝐸Δ𝑡 ≥ ℎ, which governs the time


integration of the quantum wave equations: near initial timethe energy is very broad and narrows with increasing timeto give a spectrum of energy levels (Figure 3). Quantumdynamics, however, does share with classical dynamics theproperty that stationary (i.e., constant energy) solutions areobtained.

The envelop of trajectory amplitudes shown in Figure 1is observed to increase over time because the quantum tra-jectories (hereafter called spectral trajectories) are calculatedusing the time-dependent solutions (hereafter called thespectral solutions [10]), which are a superposition of quantumstates including the continuum, such that the envelop growthover time reflects the excited and continuum content of thewave function. On the other hand the envelop of trajectoryamplitudes shown in Figure 4 is not observed to increaseover time because the trajectories (hereafter called eigen-trajectories) are calculated using eigenfunctions as filteredfrom the spectral solutions at a specific stationary energy asdetermined fromFigure 3, as described in [10]. In some of thecalculations Hamiltonian expectation values calculated usingeigenfunctions and eigentrajectories (Figure 5) still exhibit anoscillation in the time. In these cases the physical energiescan be found from the time average of the Hamiltonianexpectation values (Figures 5 and 6); in other words the timeaverage is sensibly stationary.

3. Quantum Trajectories in the NonrelativisticLimit

The time-independent solution of (2b) can be written as

𝜒 ( ⃗𝑟, 𝑡) =

−𝑖ℎ𝑐�⃗� ⋅ ∇⃗𝜓 ( ⃗𝑟, 𝑡)

𝐸 − 𝑉 + 𝑚𝑐2

. (11)

Evaluating 𝜒( ⃗𝑟, 𝑡) in the nonrelativistic limit, for which (𝐸 −𝑉) ≪ 𝑚𝑐

2, we immediately see that the current given by(3) is of order 𝑐0 in this limit. Remarkably the 𝑐0 order ofthe quantum trajectories is in agreement with experimentalobservation that Fermi-Dirac statistics is obeyed for allelectron velocities, including those whose Dirac current iscalculated in the regime of nonrelativistic velocity.

The trajectories are quantum mechanical since they canbe calculated either using Dirac’s equation in the regime ofrelativistic velocities or Schrodinger’s equation in the regimeof nonrelativistic velocities.The quantum trajectories are spindependent since they depend explicitly on Pauli’s vector. Ourresult, for the first time, gives a relativistic correction which isof order 𝑐0, all other relativistic corrections being of order 𝑐−2.This relativistic correction is due exclusively to the spin of theelectron and accounts for Fermi-Dirac statistics. Traditionallyrelativistic corrections have been considered to start at order𝑐−2 for atomic fine structure. Readers will appreciate theapparent paradox that we had to appeal to Dirac’s equationto explain the omnipresence of electron spin in all fermionicphenomena notwithstanding the relativistic or nonrelativisticnature of the motion.

We believe that this unforeseen Dirac correction of order𝑐0, which is of course the same order as all Schroedinger’scontributions to the Hamiltonian, is due to the Lorentz

0

1

2

3

4

5

6

−1

0 50 100 150 200𝑡 (au)

Sze (

au)

Figure 2: Quantum eigentrajectories in the z direction showinghow the two electrons of H

2correlate with increasing time in the

formation of an antibonding state. Upper: spin-up electron. Lower:spin-up electron.The eigentrajectories are calculated from (7) in thenonrelativistic limit using eigenfunctions for two parallel spin statesof the 3Σ

𝑢state of H

2.

9𝑒 + 03

8𝑒 + 03

7𝑒 + 03

6𝑒 + 03

5𝑒 + 03

4𝑒 + 03

3𝑒 + 03

2𝑒 + 03

1𝑒 + 03

0𝑒 + 00

−2 −1 0 1

𝐸 (au)

Spec

trum

(au)

Figure 3: Spectrum at𝑅 = 2.5 au for the 1Σ𝑔state of H

2. Solid: spin-

up electron. Dotted: spin-down electron.

covariance [7] of Dirac’s equation appropriate for a spin-1/2 particle, as discussed in the last section. This surmiseis supported by the fact that the Klein-Gordon relativisticequation appropriate for a spin-0 particle, although it isLorentz invariant, it fails to satisfy the equation of continuitygiven by (1) for a physically acceptable current.


0 50 100 150 200𝑡 (au)

−3

−2

−1

0

1

2

3

Sze (

au)

Figure 4: Quantum eigentrajectories in the z direction. Innercurves: 𝑅 = 1.4 au. Outer curves: 𝑅 = 3.0 au. The inner curves showhow the two electrons correlate with increasing time and eventuallyfind the region of covalent bonding located between the two protonsfixed at 𝑧 = ±0.7 au, while the outer curves show that the electronsremain in the vicinity of the separated atoms for all times. Theeigentrajectories are calculated from (7) in the nonrelativistic limitusing eigenfunctions for the up and down spin states for the 1Σ

𝑔

state of H2.

150140 160 170 180 190 200𝑡 (au)

𝐸(a

u)

−1.5

−1

−0.5

Figure 5: Expectation values of the Hamiltonian and their timeaverages for 𝑅 = 1.4 au toward the end of the time integration.Lower: 1Σ

𝑔state. Upper: 3Σ

𝑢state. These energies, especially for

the 1Σ𝑔state, have some numerical noise (high-frequency hair

superimposed on the wave oscillation), but nevertheless the timeaverages at maximum time agree well with the time-independentresults, which are −1.174 and −0.7842 for the 1Σ

𝑔and 3Σ

𝑢states,

respectively.

−1.2

−1.15

−1.1

−1.05

−1

𝐸(a

u)

150 160 170 180 190 200𝑡 (au)

Figure 6: Time average of the Hamiltonian expectation values forthe 1Σ

𝑔state. From bottom to top for 𝑅 = 1.4 au, 𝑅 = 2.5 au,

𝑅 = 3.0 au, and 𝑅 = 4.0 au, respectively. The KW energiesare, respectively, −1.174 au, −1.094 au, −1.057 au, and −1.016 au. Theagreement with KW at the larger internuclear distances is not asgood owing to smaller binding energies whose magnitude falls intothe last digit of accuracy of the calculation.

In the nonrelativistic regime of electron velocity, (2a)–(9) may be evaluated using Schroedinger’s equation, whichfollows on exactly eliminating (2b) in favor of (2a) anddropping contributions in the resulting equation for the largecomponent, 𝜓( ⃗𝑟, 𝑡), of order 𝑐−2. The current is evaluated inthe nonrelativistic limit using 𝜓( ⃗𝑟, 𝑡) = 𝜓

𝑆( ⃗𝑟, 𝑡)𝜒

𝑚𝑠

and 𝜒( ⃗𝑟, 𝑡)given by (11) in the regime (𝐸−𝑉) ≪ 𝑚𝑐2, where𝜓

𝑆( ⃗𝑟, 𝑡)obeys

the time-dependent Schroedinger’s equation as follows:

𝑖ℎ

𝜕𝜓𝑆( ⃗𝑟, 𝑡)

𝜕𝑡

= [−

ℎ2

2𝑚

∇2+ 𝑉 ( ⃗𝑟, 𝑡)] 𝜓𝑆 (

⃗𝑟, 𝑡) . (12)

𝜒𝑚𝑠

has up (plus sign) or down (minus sign) magneticsubstates denoted by 𝑚

𝑠= ±1/2 (i.e., 𝛼 or 𝛽 spin states),

respectively. Written out explicitly in terms of the largecomponent the current given by (3) becomes

�⃗� ( ⃗𝑟, 𝑡) ≅

ℎ

2𝑚

[−𝑖 (𝜓∗

𝑆∇⃗𝜓𝑆− 𝜓𝑆∇⃗𝜓∗

𝑆)

+ 𝜓∗

𝑆𝜒+

𝑚𝑠

(∇⃗ × �⃗�) 𝜓𝑆𝜒𝑚𝑠

−𝜓𝑆𝜒+

𝑚𝑠

(�⃗� × ∇⃗) 𝜓∗

𝑆𝜒𝑚𝑠

] ,

(13)


where we have used �⃗�+ = �⃗� and the identity, (�⃗� ⋅ �⃗�)(�⃗� ⋅ �⃗�) =�⃗�⋅�⃗�+𝑖�⃗�⋅(�⃗�×�⃗�), fromwhich the identities useful in evaluatingthe current can be inferred as follows:

�⃗� (�⃗� ⋅ ∇⃗) = ∇⃗ + 𝑖 (∇⃗ × �⃗�) , (14a)

(�⃗� ⋅ ∇⃗) �⃗� = ∇⃗ + 𝑖 (�⃗� × ∇⃗) . (14b)

Written out explicitly for up (upper sign) or down (lowersign) spin states the nonrelativistic current is

⃗𝑗𝑛𝑟( ⃗𝑟, 𝑡) =

ℎ

𝑚

[ Im𝜓∗𝑆( ⃗𝑟, 𝑡) ∇⃗𝜓𝑆 (

⃗𝑟, 𝑡)

± �̂�Re𝜓∗𝑆( ⃗𝑟, 𝑡)

𝜕

𝜕𝑦

𝜓𝑆 (

⃗𝑟, 𝑡)

∓�̂�Re𝜓∗𝑆( ⃗𝑟, 𝑡)

𝜕

𝜕𝑥

𝜓𝑆 (

⃗𝑟, 𝑡)] .

(15)

The first term on the right side of (15), which is independentof spin, is the familiar term contributed by Schroedinger’sequation, while the second and third terms, which aretransverse to the axis of quantization along z, are contributeduniquely by Dirac theory. Notice that all contributions to thecurrent are of order 𝑐0, as discussed earlier in this section.

4. Calculations and Results

The time-dependent Schroedinger’s equation [(12)] is solvedin the time and three Cartesian coordinates for each electronof H2using an algorithm described previously [11]. (Prelim-

inary He-atom calculations were presented previously [12].)The computational grid box is square with a uniform meshof 323. The accuracy of the calculation could be improved bymesh refinement of the uniformmesh or by use of an adaptivemesh (which we do not have at our disposal); however, wefind that for 𝑍 = 1 or 𝑍 = 2 the present mesh is adequate,especially for the proof of principle calculations presentedhere. The two Schroedinger’s equations were integrated inthe time for a length of 200 au using 4000 mesh points (forinternuclear distance 𝑅 = 1.4 au calculations) or 8000 meshpoints (for all other internuclear distance calculations). Atinitial time a Slater-type orbital with form 𝑟𝑒−𝜁𝑟, 𝜁 = 1/2centered on each proton is used for each electron for the 1Σ

𝑔

state and with forms 𝑟𝑒−𝜁𝑟 𝜁 = 1 and 𝑟2𝑒−𝜁𝑟, 𝜁 = 1/2 forthe electrons of the 3Σ

𝑢state. The Pauli principle is enforced

at initial time for electrons in the same spin state, and theelectrons obey the Pauli principle for all subsequent times dueto their mutual correlation. If the Pauli principle is violated atinitial time by assigning the same spatial orbital to electronsin the same spin state, then the electrons fail to correlate forsubsequent times.

The time-dependent wave function for each electron isevolved, and its spectrum of eigenvalues and eigenfunctionsis obtained using the methods described in the pioneeringpaper of Feit et al. [10] on the spectral solution of the time-dependent Schroedinger equation.The results are sensitive to

initial conditions only in the sense that a poor choice (use of𝑟0 and larger values of 𝜁, e.g.) was found to lead to trajectoryrun away and an unphysical solution. As in classical dynamicscare should be taken to use energy-conserving numericalmethods to integrate the trajectory fields using (6). Trajectoryrun away is easily recognized from a plot of trajectory versustime in which the trajectory is not periodic and grows withtime, representing an unbound electron (self ionization) orin which it is periodic at large distances from the nuclei,representing a spurious excited state (self excitation). Theenergies of runaway trajectories are observed to lie below thetrue energies due to insufficient strength of the interelectronicpotential. When sufficient care is taken to obtain energy-conserving trajectories, calculations tend to converge frombelow rather than from above experimental energies.

Readers should be reminded that the evolution of thewave function by solving (12) is not an initial-value problem,although it may appear to be from the methodology justoutlined. This is a point of confusion for workers whoseexperience is in time-independent structure theory. In factthe Heisenberg uncertainty principle, Δ𝐸Δ𝑡 ≥ ℎ, guaranteesthat at initial time the spectral width of a line is infinitelybroad and narrows with the length of the time integration,whose measure is Δ𝑡. The positions of the spectral peaks(Figure 3) are accurately predicted even for early times andwide spectral lines, as measured by Δ𝐸, but the integrationshould be continued for long enough times that the spectralpeaks for different eigenvalues are well separated. This maybe hard to accomplish for integrating (12) for heavy particles.

Figure 1 shows quantum trajectories in the y directionfor up and down spin states and identical spatial orbitalsat initial time. All calculations are performed for protonslocated at ±𝑅/2 along the 𝑧 axis. These are trajectorieswhich belong to the 1Σ

𝑔state. It is easy to see that the

two opposite-spin trajectories are correlated in the sensethat they keep apart and on opposite sides of the centers ofattraction most of the time. Notice the expanding envelopsof the amplitudes with increasing time, which indicates theincreasing excited and continuum-orbital contribution to thewave functionswith increasing time.This behavior is possiblebecause the time-dependent or spectral wave functionswhichare used to calculate the currents, whence the trajectoriesare a superposition of the complete set of states of theHamiltonians, as emphasized in [10].

Figure 3 shows the spectra at an internuclear distance of𝑅 = 2.5 au as calculated from the temporal Fourier transformof the overlaps of the wave functions at time t with the initial-time wave functions, as discussed in [10]. Having located thespectral energies at the positions of the peaks, eigenfunctions(Figures 7 and 8) are calculated from the Fourier transformof the spectral wave function precisely at the spectral peakpositions. A window function [10],𝑤(𝑡) = (1−cos(2𝜋𝑡/𝑡max),is used in both the eigenvalue and eigenfunction calculationsin order to suppress side-lobe numerical artifacts.

Figure 7 shows a z-slice at 𝑥 = 𝑦 = 0 of the eigen-functions for 𝑅 = 1.4 au, and Figure 8 similarly shows theeigenfunctions for 𝑅 = 2.5 au. The fixed locations of theprotons are clearly depicted in both plots as the twin peaks


0

0.1

0.2

0.3

0.4

WF

(au)

−4 −3 −2 −1 0 1 2 3 4𝑧 (au)

Figure 7: Wave functions along z at 𝑥 = 𝑦 = 0 at 𝑅 = 1.4 au forthe 1Σ

𝑔state of H

2. Solid: spin-up electron. Dotted: spin-down

electron. Convergence is shown for the solid curves at 0.75 𝑡max(lower) and for 𝑡max (upper) and for the upper curves at 0.75 𝑡max(upper) and for 𝑡max (lower), where 𝑡max = 200 au.

of the eigenfunctions. The nascent separation into H disso-ciation products is evident in Figure 8 by the dissymmetryof the wave functions with respect to the nuclear centers.The wave function which is symmetric about the protoncenters has the interelectronic potential set equal to zero; thusthe correct separation into He-atom dissociation productsdepends on the correlation of the two electrons. Figure 4shows corresponding quantum trajectories in the z directioncalculated using (7) in the nonrelativistic limit, as in Figure 1,but with the spectral wave function, which is a superpositionof states, replaced by eigenfunctions obtained, as mentionedpreviously and described in [10], from the temporal Fouriertransform of the spectral wave function at a spectral eigen-value. These may be considered to be eigentrajectories, asopposed to the spectral trajectories appearing in Figure 1.Notice that at later times the eigentrajectories for 𝑅 =1.4 au are pulled into the internuclear region, reflecting thecovalent nature of the bond. In contrast the eigentrajectoriesfor 𝑅 = 2.5 au remain in the separated-atoms region for alltimes, reflecting the weak covalency of the bond for largeinternuclear separation. We regard this behavior as strongjustification for the present methodology, in which the corre-lation of the two electrons guarantees the correct dissociationproducts automatically without having to build this behaviorinto the basis set in a conventional variational calculation.

Figure 2 shows eigentrajectories for electrons having thesame spin states and different initial orbitals. These aretrajectories belonging to the 3Σ

𝑢state.

0 50

0.1

0.2

0.3

WF

(au)

−5

𝑧 (au)

Figure 8: Wave functions along z at 𝑥 = 𝑦 = 0 at 𝑅 = 3.0 aufor the 1Σ

𝑔state of H

2. Solid: spin-up electron. Dotted: spin-down

electron.The dotted curve which is symmetric about the two protonpositions is calculated for the interelectronic interaction artificiallyset equal to zero, demonstrating that the correct dissociation limitinto two hydrogen atoms, as shown by the solid and dottedcurves asymmetric about the two protons, depends critically on theinterelectronic interaction.

Notice that the trajectories are separated into a ground-state trajectory centered about one nucleus and an excited-state trajectorymoving at large distances from the two centersof attraction. Such a state is known in quantum chemistry asantibonding.

Figure 5 shows the late-time energies of the two states andthe time averages of the energies. The energies are calculatedfrom the expectation value of the two-electron Hamiltonianusing the eigenfunctions inferred from the spectral solution,as discussed previously and in [10].The interelectronic poten-tials are calculated using the eigentrajectories, as discussedpreviously, and the sum over two electrons is divided by2 in order to avoid double counting. Notice that the timedependence of the potential, as given by (8)-(9), means thatthe Hamiltonian expectation values tend to oscillate about astationary time average, as shown in Figures 5 and 6.

Finally Figure 6 shows the energy time averages forfour different internuclear distances. The agreement of thetime averages with the variational calculations of Kolos andWolniewicz [13] is good, whose energies for the bondingand antibonding states are −1.174 and −0.7842, respectively.Readers are reminded that the binding energy is only 17.45%of the total electronic energy of −1.174 for the ground state ofmolecular hydrogen.The agreement at the larger internucleardistances is not as good owing to smaller binding energieswhose magnitude falls into the last digit of accuracy of thecalculation.


5. Conclusions

One cannot extract more than three significant figures fromthe calculations at the present level of accuracy, but we areconfident that the results can be systematically improvedfor accuracy by further refinements in the grid meshes andespecially by using improved, energy-conserving numericalmethods to integrate the quantum trajectories in order toavoid run away, the underestimation of the interelectronicpotential energy, and as a consequence the overestimationof the binding energy. These problems notwithstanding, webelieve that the quantum-trajectory theory (a) establishesthe relationship between the spin of an individual electronand the Fermi-Dirac statistical behavior of an ensembleof electrons and (b) achieves the pair-wise correlation ofelectrons by evaluating Coulomb’s law as an instantaneousinteraction in the time rather than as a quantum stationary-state mean or exchange interaction.

Acknowledgments

The author is grateful to T. Scott Carman for his support ofthis work.This work was performed under the auspices of theLawrence Livermore National Security, LLC (LLNS), underContract no. DE-AC52-07NA273.

References

[1] H. M. James and A. S. Coolidge, “The ground state of thehydrogen molecule,”The Journal of Chemical Physics, vol. 1, no.12, pp. 825–835, 1933.

[2] M. E. Riley, J. M. Schulman, and J. I. Musher, “Nonsymmetricperturbation calculations of excited states of helium,” PhysicalReview A, vol. 5, no. 5, pp. 2255–2259, 1972.

[3] R. J. Bartlett and M. Musial, “Coupled-cluster theory in quan-tum chemistry,” Reviews of Modern Physics, vol. 79, no. 1, pp.291–352, 2007.

[4] D. Bohm, Physical Review A, vol. 85, p. 166, 1952.[5] P. Holland, Quantum Theory of Motion, Cambridge Press,

Cambridge, UK, 1993.[6] R. E.Wyatt,QuantumDynamics with TrajecTories: Introduction

to Quantum Hydrodynamics, Springer, New York, NY, USA,2005.

[7] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics,McGraw-Hill, New York, NY, USA, 1964.

[8] H. A. Bethe and E. E. Salpeter, QuantumMechanics of One- andTwo-Electron Atoms, Dover, New York, NY, USA, 2008.

[9] H. A. Bethe and E. E. Salpeter,QuantumMechanics of One- andTwo-Electron Atoms, Dover, New York, NY, USA, 2008.

[10] M. D. Feit, J. A. Fleck, and A. Steiger, “Solution of theSchrödinger equation by a spectral method,” Journal of Com-putational Physics, vol. 47, no. 3, pp. 412–433, 1982.

[11] M. E. Riley and B. Ritchie, “Numerical time-dependentSchrödinger description of charge-exchange collisions,” Physi-cal Review A, vol. 59, no. 5, pp. 3544–3547, 1999.

[12] B. Ritchie, “Quantum molecular dynamics,” International Jour-nal of Quantum Chemistry, vol. 111, no. 1, pp. 1–7, 2011.

[13] W. Kolos and L. Wolniewicz, “Potential—energy curves for the𝑋1Σ𝑔, 𝑏3Σ𝑢, and 𝐶1Π

𝑢states of the hydrogen molecule,” The

Journal of Chemical Physics, vol. 43, no. 7, pp. 2429–2441, 1965.

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