Application of the Dyson-type boson mapping for low-lying electron excited states in molecules

Transcription

1 Prog. Theor. Exp. Phys. 05, 063I0 ( pages DOI: 0.093/ptep/ptv068 Appliation of the Dyson-type boson mapping for low-lying eletron exited states in moleules adao Ohkido, and Makoto Takahashi Teaher-training Course, Physis, Chubu University, Aihi , Japan Department of Applied Chemistry, College of Engineering, Chubu University, Aihi , Japan sohki87533@is.hubu.a.jp Reeived Deember, 04; Revised April 4, 05; Aepted April 4, 05; Published June, In many-eletron systems, partiularly when it is the ase that the system onsists of quasipartiles having almost no interation with eah other, the approahes to onsidering a quasipartile to be a basi partile are important. We propose a bosonization method of the system in whih a quasi-partile onsists of eletron hole pairs. Low-lying eletron exited states are approximately onsidered as the olletive motion states of the quasi-partiles that orrespond to the quantized normal mode osillations, and eah of these quasi-partiles onsists of the eletron hole pairs. We onsider the fundamental model Hamiltonian, and transform this into the quasi-partile representation Hamiltonian. Then, we transform these quasi-partiles into bosons by using the Dyson-type boson mapping method. The obtained Dyson-type boson Hamiltonian enables a straightforward interpretation of the low-lying eletron exited states and failitates their alulation. To onfirm the usefulness of this method, we apply this to the π-eletrons system of the ethylene moleule.... ubjet Index D0, I47. Introdution In this paper, we propose a bosonization method of the system in whih a quasi-partile onsists of eletron hole pairs. Eletron exitation energies of low-lying eletron exited states are obtained by finding out the quantized normal mode osillations around a ground state in the moleular eletroni struture. These exited states are approximately onsidered as the olletive motion states of the quasi-partiles that orrespond to the quantized normal mode osillations, and eah of these quasipartiles onsists of the eletron hole pairs. We introdue the quasi-partile operators whih reate the low-lying eletron exited states. These introdued quasi-partile operators, however, do not satisfy the ommutation relations as bosons. Therefore, by using the Dyson-type boson mapping method, we transform these operators into those whih satisfy exatly the boson ommutation relations. We disuss how to map the operators of the fermion spae onto the Dyson-type boson spae, and obtain the Dyson-type boson representation Hamiltonian. By using this Hamiltonian, we disuss the low-lying exited states as a many-body problem of the boson system. We apply this method to the π-eletrons system of the ethylene moleule, and obtain the low-lying eletron exited state energies in the random-phase approximation (RPA. In the next step, to improve the auray of the these obtained exitation energy values, we arry out additional alulations by The Author(s 05. Published by Oxford University Press on behalf of the Physial oiety of Japan. This is an Open Aess artile distributed under the terms of the Creative Commons Attribution Liense ( whih permits unrestrited reuse, distribution, and reprodution in any medium, provided the original work is properly ited. Downloaded from on 9 Deember 07

2 PTEP 05, 063I0 using these values, and we obtain improved results. From these analytial proesses and the numerial alulation results, it was onfirmed that this method was effetive. In addition, the onrete differene of the energy state struture of the original Fermi spae and that of the mapped boson spae beame lear. This is what we must note in dealing with physial quantities in the boson spae. A large number of studies on the boson mapping method have been done sine Holstein Primakoff ] first proposed it (the HP-type boson mapping method. The other type of mapping method was proposed by Dyson,3] (the Dyson-type boson mapping method. Beause the HP-type mapping is a unitary transformation, Hermitiity of the original Hamiltonian of a system is onserved in the spae where this mapping was made. This HP-type boson representation Hamiltonian, however, has the disadvantage of infinite expansions of boson operators. The Dyson-type boson representation Hamiltonian has the disadvantage that this mapping breaks Hermitiity. This mapped Hamiltonian, however, has the advantage of finite expansions of boson operators. The disadvantage of the Dysontype mapping method, namely the problem of the diagonalization of the non-hermitian matries, has been solved by the onversion method of Takada 4] in nulear physis. We have previously reported the fundamental theory of the Dyson-type boson mapping to analyze the low-lying exited state energies in moleules 5]. We have also disussed the eletron orrelation problem of moleules by using this mapping, in whih quasi-partiles onsisting of eletron eletron pairs are transformed into bosons 6].. The basi theory of the Dyson-type boson transformation We onsider the fundamental model Hamiltonian onsisting of two eletrons in the two energy levels, whih are a ground state and an exited state. This Hamiltonian provides the base of the Hamiltonian to deal with the low-lying eletron exited states, and it an be expanded to systems having many eletrons. The moleular-orbital representation model Hamiltonian (see Appendix A whih desribes the low-lying eletron exited states system is: H = U 0 + ε + ε (γ 0 + γ ( , ( + (γ 0 + γ, ( (γ 0 γ, ( ( (γ 0 γ, ( + +, ( where ε = α β, ε = α + β. ( The operator k + (k =, reates an eletron having the spin state in the k moleular orbital, and the operator k is the annihilation operator. The orbital k = is the ground state and k = is the exited state. α, β, γ 0,andγ are the Coulomb integral, the resonane integral, the one-enter eletron-repulsion integral, and the two-enter eletron-repulsion integral, respetively. We introdue the quasi-partile operators b ρ + (the physial boson operators. Eah of these operators is a linear ombination of partile hole pair operators: b ρ + = ψ ρ (, +, (3 / Downloaded from on 9 Deember 07

3 PTEP 05, 063I0 and b + ρ g =b+ ρ } { = exited state ρ, (4 b ρ g =0, (5 where ρ indiates the type of the exited state. 0 is the vauum state and g is the ground state. The ommutation relation of the operators b ρ + and b ρ is as follows: ] b ρ, b ρ + =, ψ ρ (, ψ ρ (, { ψ ρ (, ψ ρ (, 3 3 +,, 3 + ψ ρ (, ψ ρ ( 3, + 3 }. (6 The operators b ρ + and b ρ do not satisfy the boson ommutation relation. We give the following onditions in advane to the determine the oeffiients ψ ρ (, : ψ ρ (, ψ ρ (, =, (7, ψ ρ (, ψ ρ (, = δ, δ,. (8 ρ To arry out the Dyson-type mapping, we alulate the double ommutation relation between one b ρ and two b ρ + in onsideration of Eqs. (7 and(8: ] ] b ρ, b ρ +, b ρ + = g (ρρρρ b + ρ, (9 ρ g(ρρρρ = ψ ρ (, ψ ρ (, 3 ψ ρ ( 4, ψ ρ ( 4, 3, (0, 3, 4 and the following relations about the oeffiients g(ρab are derived from the above equation: g(ρab = g(baρ = g(aρb = g(bρa. ( Considering the double ommutation relation Eq. (9, we arry out the Dyson-type mapping of the operators b + ρ and b ρ: b + ρ B+ ρ ab G(ρ abb + a B+ b B, ( b ρ B ρ, (3 where g(ρ ab + g(ρ ba G(ρ ab =. (4 The operators B ρ + and B ρ are the ideal boson operators, whih satisfy the boson ommutation relations. On the basis of the above onsideration, we transform the original fermion Hamiltonian into the Dyson-type boson Hamiltonian: H( + k, k H(b + ρ, b ρ H(B + ρ, B ρ. (5 3/ Downloaded from on 9 Deember 07

4 PTEP 05, 063I0 Fig.. Graphial representations of the physial boson operators. Table. The values of the oeffiients ψ ρ (,. ψ (, = ψ (, = 0 ψ (, = 0 ψ (, = ψ T (, = 0 ψ T (, = ψ T (, = ψ T (, = 0 ψ T (, = 0 ψ T (, = ψ T (, = ψ T (, = 0 ψ T3 (, = ψ T3 (, = 0 ψ T3 (, = 0 ψ T3 (, = 3. Appliation We onsider the four exited states ρ (ρ =, T,T,andT 3. The quasi-partile operators to reate these states are given in the form of the following four equations, and Fig. shows these states. The values of the oeffiients ψ ρ (, are given in Table. b + = ( = +, (6 b + T = ( =, ( +, (7 b T + = ( + + = b T + +, (8 b T + 3 = ( + + = b + +. (9 The oeffiients ψ ρ (, in Table satisfy the relations Eqs. (7 and(8. About these quasi-partile operators, we obtain the following double ommutation relations: ] ] b ρ, b ρ +, b ρ + = b ρ + (ρ =, T, T, and T 3. (0 Considering the above double ommutation relation Eq. (0, we derive the following mappings: b + ρ (b+ ρ D = B + ρ B+ ρ B+ ρ B ρ, ( b ρ (b ρ D = B ρ. ( In the mapped boson spae, the ommutation relation Eq. (0 is the following equation: ] ] (b ρ D,(b ρ + D,(b ρ + D = (b ρ + D. (3 We transform the fermion Hamiltonian Eq. ( into the boson representation Hamiltonian using the following proedure. 4/ Downloaded from on 9 Deember 07

5 PTEP 05, 063I0 (i The seond term ε + and the third term ε + Considering the ommutation relations ] +, b ρ + = b ρ + (4 and ] +, b+ ρ = b ρ +, (5 + and + are found to be mapped with the following relations: +, + B ρ + B ρ, (6 ρ beause, in the mapped boson spae, the following ommutation relation Eq. (7 orresponds to those of Eqs. (4 and(5: ρ B + ρ B ρ,(b ρ + D = ρ B + ρ B ρ, B ρ + B+ ρ B+ ρ B ρ = (b ρ + D. (7 Aordingly, onerning the seond and third terms of Eq. (, the mappings to the boson spae are found to be the following orresponding relation: ε + ε + (ε ε B ρ + B ρ. (8 ρ (ii The fourth term: 4 (γ 0 + γ, ( The ommutation relation k=, ( ( k + + k k k, b ρ + = k + + k k k, k= = b + ρ + b+ ρ k + + k k k, b ρ + ( (9 From Eqs. (6 and(7, and the following ommutation relation: ( B + ρ B ρ + B + ρ B + ρ B ρ B ρ, (b ρ + D = (b ρ + D + 4(b ρ + DB ρ + B ρ, (30 ρ the mapping to the boson spae of the fourth term of Eq. ( is found to be the following orresponding relation: 4 (γ 0 + γ ( , ( (γ 0 + γ ρ ( B + ρ B ρ + B + ρ B+ ρ B ρ B ρ. (3 5/ Downloaded from on 9 Deember 07

6 PTEP 05, 063I0 (iii The fifth term: (γ 0 + γ (, + + The ommutation relation ( + +, b + ρ = b ρ + b+ ρ, ( (3 As the mapping to the boson spae of the fifth term of Eq. ( is the same as that of the fourth term, the fifth term is mapped with the following orresponding relation: (γ 0 + γ ( + + (γ 0 + γ (B ρ + B ρ B ρ + B+ ρ B ρ B ρ. (33, ρ From Eqs. (3 and(33, the fourth term and the fifth term anel eah other in the mapped spae. (iv The sixth term: 4 (γ 0 γ, ( ( In onsideration of the two forms of exitation about the spin part of the operators, we rearrange the operators into the following two forms: ( (34, ( and, ( (35 ( From Eqs. (6 and(9, Eq. (34 is transformed into the physial boson form: ( ( = , ( = (b b T3 ( ( ( b + b T3 + b + b+ T 3 b + + b+ T 3. (36 From Eqs. (7 and(8, in the same way, Eq. (35 is transformed as follows: ( = b T + b T b + T + b + T. (37, ( From Eqs. (36 and(37, the sixth term is mapped with the following orresponding relation: 4 (γ 0 γ, ( ( ( ( 4 (γ 0 γ b b D D {( ( + ( i b T + + i b Ti D D} ]. (38 i= 6/ Downloaded from on 9 Deember 07

7 PTEP 05, 063I0 (v The seventh term: (γ 0 γ, ( + + ( + + =, = + +, b + b. (39 AordingtoEq.(39, the seventh term is mapped with the following orresponding relation: (γ 0 γ { ( + + (γ 0 γ } B ρ + B ρ + (b + D (b D. (40, ρ From these proedures, the Dyson-type boson representation Hamiltonian H D is obtained as follows Eqs. (8, (38, and (40]: ( H D = (α + β β + γ { 0 γ B + B + + γ 0 γ 4 ( b + } 3 B T + i B Ti + (γ 0 γ ( b + D (b D i= (b 3 {( ( + + ( i b + D D T + i b Ti D D} ]. (4 4. The exitation energy alulation for ethylene 4.. The exitation energy alulation in the RPA As a onrete appliation, we onsider the ethylene moleule having two π-eletrons in the two energy levels. To obtain the exitation energies, we onsider the ommutation relations with the operators B + ρ, B ρ and the Hamiltonian H D Eq. (4]. However, beause of the ompliated produts of the boson operators in this Hamiltonian H D, we are faed with diffiulty in obtaining the energy eigenvalues. Dealing with this diffiulty, we perform the RPA on the Hamiltonian H D.TheRPAin this ase means that the operator (b ρ + D is replaed with the operator B ρ + : We define H D under RPA as H RPA D i= (b + ρ D B + ρ. (4 (the RPA Hamiltonian: { } HD RPA = (α + β + A B+ B + A B + + B + where Γ = γ 0 γ, A = (β Γ, A = Γ, 3 i= A T i = (β + Γ, AT i = Γ ( i, (i =,, 3. { }] A T i B+ T i B Ti + A T i B T + i + BTi, (43 7/ Downloaded from on 9 Deember 07

8 PTEP 05, 063I0 Table. Integral values (ev. Parr 0] Ohno Klopman,] γ γ β Table 3. Our alulations under the RPA for ethylene (ev. Calulation (a Calulation (b Experimental ω (singlet ω T (triplet ω Ti (i =, ω ω T For the integral values γ 0, γ, and β, we used the values of Parr in Table. For the integral values γ 0, γ, and β, we used the values of Ohno Klopman in Table. H RPA D follows: satisfies the Hermitian property. The ommutation relations with B + ρ, B ρ,andh RPA D are as ] HD RPA, B+ ρ = A ρ B+ ρ + Aρ B ρ, (44 ] HD RPA, B ρ = A ρ B+ ρ Aρ B ρ. (45 The exitation energies ω ρ in the RPA are obtained by solving the seular equation of Eqs. (44 and (45: ω ρ = (A ρ 4(A ρ (ρ =, T, T, T 3. (46 For the integral values β, γ 0,andγ, we use the two kinds of values whih were alulated by Pariser and Parr (semi-empirial 7 0] and Ohno and Klopman (semi-empirial,]. These integral values are shown in Table. The results of our alulations under the RPA for the exitation energies ω ρ (ρ =, T, T, T 3 of ethylene are given in Table 3. The exitation energy ω is the singlet state and ω T is the triplet state. In our alulation at this point, the differenes between the energies ω T and ω Ti (i =, 3 annot be distinguished. However, as seen from Eqs. (7and(9 and Fig., it is neessary to note the point that the energy states ω Ti (i =, 3 do not exist in the original Fermi spae. For ω T3, both the orbital part and the spin part of the eigenfuntion onerned are anti-symmetri in the interhange of the oordinates of the two eletrons. Therefore, this eigenfuntion of ω T3 does not have the antisymmetri harater. For ω T, the exited state orresponding to this type does not exist from the first in the Fermi spae. 4.. The exitation energy alulation starting with the energies obtained in the RPA We transform the operators B ρ + and B ρ into the operators R ρ + and R ρ,wherer ρ + is the operator to reate the eigenstate in the RPA diretly and R ρ is its omplex onjugate; they satisfy the following equations: ] HD RPA, R+ ρ = ω ρ R ρ +, Rρ, R ρ + ] =, (47 8/ Downloaded from on 9 Deember 07

9 PTEP 05, 063I0 Table 4. The values of the transformation oeffiients ρ and ρ. Calulation (a Calulation (b T i (i =,, (i =,, T i where ω ρ are the exitation energies in the RPA Eq. (46]. From these equations, we obtain the following relations: R ρ + = ρ B+ ρ + ρ B ρ, R ρ = ρ B ρ + ρ B+ ρ, (48 where ρ and ρ are the transformation oeffiients, the values of whih are shown in Table 4. The Hamiltonian Eq. (4] expressed in the operators R ρ + and R ρ is as follows: where h RPA H D ( R + ρ, R ρ = h RPA 0 + h 0, (49 0 ={onstants}+w ρ R+ ρ R ρ + W ρ ( R + ρ ρ + W 3 R ρ, (50 { h 0 = the terms exept those whih ontain the operators R ρ + R ρ, ( R ρ + } and R ρ. (5 With respet to the terms W ρ j ( j =,, 3, for example, the values W, W,andW 3 are as follows: { ( ( } ( { ( ( } W = (β Γ + 4Γ + { ( ( } ( { 3 ( ( } 3 Γ Γ Γ 7 + 8, { ( ( } { ( ( } ( ( W = (β Γ + Γ Γ + 3 Γ ( { ( ( } + 8 ( Γ , { ( ( } { ( ( } W3 = (β Γ + Γ Γ + { ( ( } ( 6 ( ( { ( ( }] 3 4 ( Γ Γ (5 The Hamiltonian h RPA 0 does not have Hermitiity beause W ρ W ρ 3. For any fermion Hamiltonian H, we express its HP-type boson mapping Hamiltonian as H HP and its Dyson-type boson mapping Hamiltonian as H D. The matrix elements of these Hamiltonians H HP 9/ Downloaded from on 9 Deember 07

10 PTEP 05, 063I0 Table 5. Our alulations for ethylene obtained by using the RPA eigenstates (ev. Calulation (a Calulation (b Parr Ohno Klopman Experimental ω (R ρ +, R ρ (singlet ω T (R ρ +, R ρ (triplet ω Ti (R ρ +, R ρ(i =, ω ω T and H D satisfy the following relations 4]: { (i (i H HP i = L (i H D i R = (i H D i H D i } / (i H D i, (53 where i represent the eigenvetors of the H HP,and L (i and i R represent the left-hand-side eigenvetors of H D and the right-hand-side eigenvetors of H D, respetively. To alulate the eletron exitation energies of the Hamiltonian h RPA 0 in aordane with Eq. (53, we hoose ρ RPA = R ρ + g RPA as the eigenvetors g RPA is the RPA ground state]. Beause these are the eigenvetors of HD RPA having Hermitiity, and an also be appliable as the eigenvetors of the HP-type boson mapping Hamiltonian. For these eigenvetors, we alulate the following ommutation relations: R ρ +, R ρ +, ] hrpa 0 hrpa 0, R ρ, ], R ρ, ] h RPA 0, h RPA 0 ]. (54 After making similar alulations for the ase of et. 4., we obtain the improved eletron exitation energies ω ρ (R + ρ, R ρ shownintable5. 5. Conlusion In the present paper, we have disussed how the quasi-partile operators are mapped onto the Dysontype boson spae in order to disuss the eletron exitation energies of the low-lying exited states in moleules. We first onsidered the system onsisting of two eletrons in the two energy levels and presented the orresponding moleular-orbital desription model Hamiltonian Eq. (]. We also obtained the Dyson-type boson representation Hamiltonian Eq. (4] from this Hamiltonian by the theoretial proedures of the Dyson-type mapping method. This Hamiltonian forms the basis for disussing the low-lying eletron exitation problem, and it an also be expanded for the ase of systems having many eletrons. We applied this method to the π-eletrons system of the ethylene moleule. This system onsists of two eletrons in the two energy levels, and it orresponds to the fundamental Hamiltonian. This appliation is appropriate to understand the struture of the mapped spae. We obtained the low-lying eletron exitation energies of this system in the RPA. Despite the simple approximation suh as the RPA, the obtained numerial alulation results agree well with the experimental results (Table 3. In this step, however, the differene between ω T and ω Ti (i =, 3 ould not be distinguished. The states ω Ti (i =, 3 annot exist in the original Fermi spae and belong to the non-physial part in the boson spae. 0/ Downloaded from on 9 Deember 07

11 PTEP 05, 063I0 In order to improve the auray of the obtained exitation energies, we transformed the operators B + ρ, B ρ into the operators R ρ +, R ρ.asshownintable5, the alulated values improved reasonably, and in this alulation the differene between ω T and ω Ti (i =, 3 was made lear. The onrete treatment of a non-hermitian Hamiltonian was given in Eq. (54. These results suggest that the Dyson-type boson mapping is useful for investigating the low-lying eletron exited states in moleules. Aknowledgment The authors would like to thank the late Dr Tanimoto for suggesting this problem and stimulating interest in it. Appendix To onsider the eletron exited state system onsisting of two eletrons in the two energy levels, we start with the following PPP Hamiltonian 7 9], whih is expressed in the atomi orbitals: H = α ( a + a + a + a + β ( a + a + a + a + γ 0 (n n + n n, ( + γ, (a + a+ a a + a + a+ a a, (A where n k = a k + a k (k =,. The fermion operator a k + reates an eletron having the spin state in the k orbital, and the operator a k is the annihilation operator. α, β, γ 0,andγ are the Coulomb integral, the resonane integral, the one-enter eletron-repulsion integral, and the two-enter eletron-repulsion integral, respetively. We transform the atomi-orbital representation Hamiltonian into the moleular-orbital representation Hamiltonian by use of the operators k +, k (k =, : + = (a + + a+ + = (a + a+, (A. (A3 Referenes ] T. Holstein and H. Primakoff, Phys. Rev. 58, 098 (940. ] F. J. Dyson, Phys. Rev. 0, 7 (956. 3] F. J. Dyson, Phys. Rev. 0, 30 (956. 4] K. Takada, Phys. Rev. C 38, 450 (988. 5]. Ohkido and O. Tanimoto, J. Mol. trut. (Theohem, 3, 43 (99. 6]. Ohkido and O. Tanimoto, Int. J. Quantum Chem. 8, (980. 7] R. Pariser and R. G. Parr, J. Chem. Phys., 466 (953. 8] R. Pariser and R. G. Parr, J. Chem. Phys., 767 (953. 9] J. A. Pople, Trans. Farad. o. 49, 375 (953. 0] R. G. Parr, The Quantum Theory of Moleular Eletroni truture (W.A. Benjamin, New York, 963, p. 59. ] K. Ohno, Theoret. Chim. Ata, 9 (964. ] G. Klopman, J. Am. Chem. o. 86, 4550 (964. / Downloaded from on 9 Deember 07