Spin-Unrestricted Multi-Conf i guration Time-Dependent Hartree Fock Theory

Wen-liang Li,Ji-cheng Bian,Lei Yang

Key Laboratory at Universities of Education Department of Xinjiang Uygur Autonomous Region for New Energy Materials,Xinjiang Institute of Engineering,Urumqi 830091,China

(Dated:Received on October 29,2013;Accepted on March 12,2014)

Spin-Unrestricted Multi-Conf i guration Time-Dependent Hartree Fock Theory

Wen-liang Li?,Ji-cheng Bian,Lei Yang

Key Laboratory at Universities of Education Department of Xinjiang Uygur Autonomous Region for New Energy Materials,Xinjiang Institute of Engineering,Urumqi 830091,China

(Dated:Received on October 29,2013;Accepted on March 12,2014)

Based on spin-unrestricted hartree fock theory,we present the spin unrestricted multiconf i guration time dependent hartree fock theory(UMCTDHF)to describe the electron correlation dynamics of systems interacting with laser f i eld.The positive spin orbitals and the negative spin orbitals are propagated in their own subspace respectively.The spin orbital in the spin-down subspace acts with that in the spin-up subspace by the reduced density matrix and mean f i eld operator.The ground energy is acquired by propagating the trial wave function in the imaginary time by using spin-restricted MCTDHF(RMCTDHF)and UMCTDHF respectively.Then the ionization probabilities and the electrons energies are calculated by using RMCTDHF and UMCTDHF when the laser f i eld is present.The ionization probability calculated with UMCTDHF agrees with the previous theoretical reports very well.The UMCTDHF method is accurate and applicable for open shell system beyond the capability of the RMCTDHF method.

Multi-conf i guration time dependent hartree fock theory,Electron-electron correlated,Strong laser f i eld,Spin-unrestricted

I.INTRODUCTION

Over recent years,there has been increasing interest in the correlated dynamics of many-electrons systems [1-5].A variety of explicit time-dependent versions of electronic structure methods have been developed,such as an important method known as multi-conf i guration time-dependent Hartree-Fock(MCTDHF)[6-13].In 2003,Zanghellini et al.developed the MCTDHF method to deal with multi-electron dynamics in strong laser f i elds[6].They showed that the MCTDHF method provided a good approximation of time-dependent multi-electron wave-functions[7]and was used to deal with the correlation of multi-electron system in strong laser f i elds[8].In 2004,Kato et al.developed the theory in the second-quantization formalism[9,10].In 2005, Nest et al.developed the MCTDHF method by using atomic basis functions[11-14],following which they studied the electronically excited states of molecules [15]and ultrafast electron dynamics in LiH[16,17] by using the MCTDHF method.The principle dif f erence between the two versions is the dif f erence in the type of basis function.Nest et al.used atomic basis functions in their theory[11-14]but Zanghellini et al. used grid vectors as the basis functions[6-8].Recently, Birkeland et al.investigated the impact of the electronelectron correlation on the ionization dynamics of Helium in intense laser f i elds by solving time-dependent Schr¨odinger equation[18].And Hochstuhl et al.studied the two-photo ionization of Helium with MCTDHF [19].Recently,the time dependent ionization probabilities of Helium are calculated[20,21].The spin function is not explicit in the wave function and working equations.The spatial orbital of the spin-down function is the same as that of the spin-up function throughout the time when the wavefunction is propagated.We call the f i rst option the RMCTDHF(R:restricted-spins) method.When the spatial orbital of the spin-down function is dif f erent from that of the spin-up function, we call that spin-unrestricted multi-conf i guration timedependent Hartree Fock(UMCTDHF)method.In order to develop UMCTDHF method,two new sets of equations are derived exactly,and the spin orbital is divided into spin-down spaces and spin-up spaces.Each spin orbital is propagated in its own space.

In this work,we describes brief l y the theory of UMCTDHF and demonstrates the test calculation on the helium atom.The results of RMCTDHF are also presented for comparison.

II.THEORY

We present an outline of UMCTDHF.Our notation follows closely the standard notation of the MCTDHmethod for nuclear quantum dynamics[20-24].

In UMCTDHF,the N-electrons wave-function can be linearly combined using time-dependent Slater determinants.

The capital letter J is a composite index which enumerates the N spin orbitals appearing in the determinant, and xi=(ri,si)is a composite variable for the position riand the spin si.In the above equation the linear coefficients AJand the spin orbitals χj1(x1,t)are all assumed to be time-dependent,and|χ1χ2χ3i is an unrestricted Slater determinant.

The Hamiltonian describing the N electron system which interacts with the strong laser f i eld:

The external potential V(ri)is produced by the(f i xed) nuclei Vnuc(ri)and may also contain the coupling to an additional laser f i eld Vexter(ri).With the aid of the variation principle one f i nds the following when varying the coefficients:

Variation with respect to the spin orbital χi1(x1,t) yields

where P is the projector operator on the total spin space spanned by all the spin orbital functions,ρ is the density operator,and hHiijis the so-called mean f i eld operator.

We can obtain the motion equations of time dependent spatial orbital in α and β space respectively, It is important to emphasize here that the reduced density matrix and the mean f i eld operator are also constructed in the total spin orbital spaces.Consequently, the two subspaces act with each other through the reduced density matrix and the mean f i eld operator linking the spin-down subspace and spin-up space.Eqs. (3),(8)and(9)are the working equations of the UMCTDHF theory.These are similar to the relationships between the restricted spin Hartree Fock method and the unrestricted spin Hartree Fock(HF)method[25]. When the spatial orbital for the spin-down and spin-up subspaces are constrained to be the same,two sets of working Eq.(8)and Eq.(9)are reduced to one set equation.

III.RESULTS AND DISCUSSION

In order to test the validity of the UMCTDHF method,the imaginary time propagation and the real time propagation have been performed respectively.For simplicity,we take helium atom for example.Firstly,we calculate the ground state energy of the helium atom by propagating the initial wave function in imaginary time in the absence of the external laser f i eld.

All our calculations have been done by using the orthogonal basis set.For a given number of spin orbitals M,there are CNMindependent determinants.We employ the usual CASSCF notation CAS(N,M)to refer to a system with N electron and M spatial orbital for spin-down and spin-up space.The spin orbitals of spindown and spin-up subspace are orthonormal to each other.However,the spatial orbital in the spin-down space is not orthonormal to that in the spin-up space.

FIG.1 Energy expectation values for the ground state of the helium atom calculated by using(a)RMCTDHF and(b) UMCTDHF.

FIG.2 The conf i guration components|A2|are plotted as a function of time.The results with CAS(2,2)are taken for example.(a)RMCTDHF,and(b)UMCTDHF.1 represent the f i rst spin orbital χ1,2 represent the second spin orbital χ2, and 12 represent the conf i guration component χ1χ2,and so forth.

The initial coefficients AJare set to N-1/2sfor all possible unrestricted Slater determinants.The Nsis the number of Slater determinants.The initial spatial orbital was achieved by taking a Hartree Fock calculation.The HF calculations have been performed by using Molpro[26]or SMILES2007[27,28]basis set of avdz.The relaxation energies during the imaginary propagation time are plotted as a function of the propagation time as presented in Fig.1.By using RMCTDHF and UMCTDHF methods,we have increased the active space from 1 spatial orbital to 6 spatial orbitals.It is obvious that both of the results calculated by RMCTDHF with CAS(2,1)and that calculated by UMCTDHF with the same active space can reproduce the HF ground state energy perfectly.And the results of CAS(2,2),CAS(2,3),and CAS(2,6)for both methods are also shown.It can be found that with the increase of the spatial orbital,the results become much closer to the full conf i guration interaction energy(FCI)calculated by Molpro[24].The relax energy of PIT with the number of 6 spatial orbitals is 2.8894 Hartree which is nearly equal to the FCI enegy 2.8895 Hartree of helium atom calculated by Molpro [24].By comparing the calculated results of the RMCTDHF and UMCTDHF with the same active spaces,we can f i nd that the relax energy can reach the same energy value.But the relaxation processes of the RMCTDHF are dif f erent from that of UMCTDHF by propagating the same trial wavefucntion.We take CAS(2,2)for example.There are four spin orbitals χ1=?1α,χ2=?2β, χ3=?3α,and χ4=?4β in the calculations.For RMCTDHF,the spatial part of spin orbital χ1(χ3)is equal to that of spin orbital χ2(χ4).For UMCTDHF,the spatial parts are dif f erent from each other.There are six conf i gurations χ1χ2,χ1χ3,χ1χ4,χ2χ3,χ2χ4,and χ3χ4available.The relaxed energy maintains steadily for a period times,and then relaxes the real ground state for RMCTDHF.For UMCTDHF,the energy relaxes the real ground state directly.In order to investigate in detail the underlying fundamental reasons for the different processes,the conf i guration components which are calculated by projecting the time-dependent wavefunction onto the available conf i gurations are plotted as a function of the propagation time in Fig.2.From the results of the RMCTDHF(Fig.2(a)),it is clearly seen that the conf i guration components converge after about10 a.u..Then the component maintains steadily for a period of time,then varies at about 25 a.u.In addition to the relaxation process dif f erence of RMCTDHF and UMCTDHF,the conf i guration components are also different from each other.When relaxation process ends, the conf i guration components are illustrated in Fig.3 for both RMCTDHF and UMCTDHF.The maximum values of conf i guration components in RMCTDHF calculations are 0.9552,0.96213,and 0.93096 for CAS(2,2), CAS(2,3)and CAS(2,6)respectively.And while the maximum one calculated by UMCTDHF are 0.9279, 0.9552,0.88885 respectively for CAS(2,2),CAS(2,3), CAS(2,6)which have 6,15,66 conf i gurations respectively.It is obvious that the conf i guration component of UMCTDHF is dif f erent from that of RMCTDHF.

FIG.3 The constitute of conf i guration components of RMCTDHF and UMCTDHF after the imaginary time propagation.(a)CAS(2,2),(b)CAS(2,3),and(c)CAS(2,6).

FIG.4 The ionization probability are plotted as a function of time.(a)avdz and(b)6-311G(2df,2pd).

The real time propagation has been performed after the imaginary time propagation.The beginning wave function used in real time propagation is the ground state wave function with the CAS(2,6)for both RMCTDHF and UMCTDHF.The helium atom,which is prepared in the ground state,is exposed to a short,intense attosecond laser pulse.The f i eld is linearly polarized and has a sine-squared temporal prof i le with ω=5 a.u. In the calculation,we take the same laser f i eld as in Ref.[18].We can calculate the ionization probability

where Φiis the bound state of the helium atom.By projecting the time-dependent wavefunction onto the bound states,the probability in the bound state can be achieved in the calculation.In Fig.4,we plot ionization probabilities as a function of propagation time for six-cycle pulses at laser frequency ω=5 a.u.by using basis set of avdz and 6-311G(2df,2pd)respectively.In Fig.4(a)the ionization probability calculated by UMCTDHF and RMCTOHF methods in di ff erent fi eld strength,i.e.Emax=0.01 and 0.05 a.u.are given. It is obvious that the calculated results of RMCTDHF are lower than that of UMCTDHF for both laser fi eld Emax=0.01 and 0.05 a.u.In Fig.4(b),only the results of UMCTDHF are presented,due to the similar trend to Fig.4(a).The ionization probabilities from Ref.[18]are shown by the points.It can be seen that the ionization probabilities used by basis set of 6-311G(2df,2pd)agree with the results taken from Ref.[18]very well.By comparing Fig.4(a)and(b),we can get that the ionization probability calculated by RMCTDHF and UMCTDHF by using basis set of 6-311G(2df,2pd)are both higher than the results by using basis set of avdz.And the ionization probability with basis set of avdz is lower than the results from Ref.[18].It may be due to that the number of basis set of 6-311G(2df,2pd)is bigger than that of avdz,it can describe well the time-dependent system wave-function when the laser fi led is presented.

FIG.5 The electron energies are calculated when the laser fi eld with di ff erent maximum fi eld strength is presented. (a)E=0.01 a.u.(b)E=0.05 a.u.(c)Laser fi eld.

FIG.6 The conf i guration components χ1χ2and χ1χ4are plotted as the real time when the laser f i eld are present. (a)The laser f i eld with Emax=0.01 a.u.,(b)and(c)are the conf i guration component for χ1χ2and χ1χ4respectively, solid line:UMCTDHF and dotted line:RMCTDHF.

The energies of the electrons are plotted as a function of the time when laser f i eld is present.As shown in Fig.5,it is obvious that the time-dependent electron energies with Emax=0.05 a.u.is higher than that of Emax=0.01 a.u.The electron energies calculated by RMCTDHF and UMCTDHF are nearly the same with the same laser f i eld.When the laser f i eld breaks of f,the electron energy with UMCTDHF is lower than that of RMCTDHF.Both processes of RMCTDHF and UMCTDHF are nonadiabatic processes[29-31],the part ground state wavefunction goes to the excited state. The state of the interaction process is a mixed state with many components.We also want to know how the conf i guration component varies when the laser f i eld varies.In Fig.6,the conf i guration component χ1χ2and χ1χ4are plotted as a function of the time.Figure 6(a)is the laser f i led whose maximum laser strength Emax=0.01 a.u.From Fig.6(b),we can see that the time-dependent conf i guration component of χ1χ2in RMCTDHF is dif f erent from that of UMCTDHF.Figure 6(c)shows the variation of the conf i guration component χ1χ4.Both the conf i guration components of RMCTDHF and UMCTHDF display the same trends.

IV.CONCLUSION

An extended theoretical method based on unrestricted spin Hartree Fock theory,named UMCTDHF, is presented in this work.The total spin orbital space is divided into two subspaces(spin-down and spin-up subspaces).The time-dependent spin orbitals have been propagated in the two subspaces independently.The spin orbital in the spin-down subspace acts with the spin oribitals in the spin-up subspace by the common reduced density matrix and mean f i eld operator,which are constructed in the total spin orbital space.Some illustrative calculations have been done.The ionization probabilities calculated by UMCTDHF agree with the latest theoretical results quantitatively.The UMCTDHF method is accurate and applicable for open shell system beyond the capability of the RMCTDHF method.

V.ACKNOWLEDGMENTS

This work was supported by the Scientif i c Research Program of the Higher Education Institution of Xinjiang,China(No.XJEDU2013S45).

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?Author to whom correspondence should be addressed.E-mail：wenliangli.dicp@gmail.com