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Calculation of vertical ionization potentials using the one-body green's function: Ne, Mg and H2O
Volume 47, number 2
CALCULATION OF VFlTICAL
CHEMICAL PHYSICS LETTERS
15 April 1977
IONIZATION POTENTIALS
USING THE ONBBOL 1(GREEN’S FUNCTION: NE, Mg, and H,O H. YAMAKAWA, T. AOYAMA Department of Chemistry, Faculty of Science, the Universt.:) of Tokyo, Hongo. Bunkyo-ku. Tokyo, Japan
and H. ICHIKAWA Division of ChemicalPhysics, Hoshi Collqe cf Pharmacy, Z-4 Ebara, Shinagawa-ku, Tokyo, Japan
Received 6 September 1976 Revised manuscript received 5 November 1976
The valence and core ionization potentials of Ne, Mg, and Hz0 have been calculated by using the one-body Green’s function. The self-energy part WBStaken to include the second order and the w2dcpcndent terms, third order ita the per-
turbation.
1. Introduction Recent development of photoelectron spectroscopy has made it an indispensable tool for establishing and clarifying orbital schemes of organic and inorganic molecules. The Hartree-Fock approximation (HF) and Koopm.ns’ theorem are usually applied to the analysis of the photoelectron spectra. However, it is well known that calculations based on the HF wavefunction and on Koopmans’ theorem give values iarger than those of experiment. ‘?ne difference between the calculated and experimental values (Koopmans’ defects) turn out to be several eV in the case of valence ionization potentials and over 10 eV in the case of core ionization potentials. A theoretical method must therefore remove the Koopman’s defect. Shtce the sudden approximation [l] may hold in photoelectron spectroscopy’, the ionization potentials (A&,, j may be expressed as AEon = E(N- 1, n) - E(N, 0) , where E(N, 0) is the energy of the ground state of the initial state of the N-partick system, and Et/V-I, n) is the energy of the nth state of the final state [(/V-l)-particle system]. Several approximate methods [2-6] which evaluate the energy difference between these two states have beer presented. Two methods are based on the application of many-body perturbation theory (5,G]. This utilises the property of the poles of the one-body Green’s function being the ionization potentials and the electron afficiti?s Some formalisms which consist of different approximations for the self-energy part Z(w) have been presented t!, Doll and Reinhardt [5! and Cederbaum and co-workers [f&12]. Their results compare weil with the erperimenmi values of rhe valence ionization potentials, but so far no emphasis has been given to core icnizntion potentials. ft is known that the single-particle-excitation approxtmation does not apply to the case of core-ionir*jtion, since intcnsc satellite lines are observed together with the principal line [ 131. The satellite lines are not calculated here by usiris
CHEMICAL
Volume 47. number 2
PHYSICS LETTERS
15 April 1977
the theory of the one-body Green’s function. We believe however that the principal lines may be calculated in this way. The principal lines have here been caiculated by using another theoretical method within the single-particleexcitation approximation [ 143. We have calculated the valence and core ionization potentials (the principal lines) of Ne, Mg, HzO, and compare them with the experimental data.
2. Theory The hamiltonian H of the molecular system is, H=
~h,+r~-“21~i-~+-‘,
hi = -(A2/2m)ViZ + Ce2 4
Ir, - ‘;-I-l ,
(1)
where i, j, and o[ represent the electrons and nuclei respectively. The behavior of the electrons in the molecular system is deduced by solving the following Schrbdinger equation or eigenvalue problem: H#
=iliagi/at,
Hq = Ep,
(2)
where 9 is usually represented by a Slater determinant to account for the Pauli principle. The practical method in which the above concept may be satisfied, the Hartree-Fock method, however, has the drawback that the interactions between
electrons
are not appropriately
included.
As is well known,
it is important
that electron
correla-
tion should be included in a transparent form. In second quantized form, the physical quantities are represented by the creation 9*(r) and annihilation G(r) field operators. The hamiltonian is H=J*JI*(r)h(r)rCl(r)r)
+ ~~~‘JI*(r)lL*(r’)(e2/1,-
r’l)$(r’)lC
(r) ,
(3)
where 9 *, 3/ satisfy the following relationships: U(r), 3/?‘)3
U(r), $@‘)I = C@*(r), $*@‘)I =‘O ,
= 6 (r-r’),
(4)
where { } is the anticommutator, {A, B} = AB + BA. The hatniltonian of eq. (3) does not contain an explicit number of particles. Therefore, such an expression for the hamiltonian is more general than that of eq. (I), since a change in the number of particles is included naturally. Thus the direct calculation of a physical quantity such as an ionization potential, which arises from a change in the number of particles, is possible by the hamiltonian expressed by eq. (3). Here we use the Heisenberg representation for convenience: ifi a+(r, t)lat = ]9(r, t),rrl ,
(5)
where [ ] is the commutator, [A, B] = AB - BA. The ground state of the IV-particle system is represented by ]lv, O>,and the one-body Green’s function is defined by the equation: G(rt,r’f’)=i-‘UV,OIT
[~,!~(r,t).$*(r’. r’)] IN, 01,
(6)
where T is Wick’s time-ordering operator. Eqs. (5) and (6) lead to the equation of motion for the Green’s function {i a/at -h(r))
G(rt, r’t’) = 6 (t - t’)6 (r - r’) +~dt”dr”Z(rt,
where the operator Z(rt, r’t’), the selfenergy, w by the Fourier transform over (t - t’) {o - h(r)) G(r, r’, w) -&r”Z(r,
r”t”)G(r”t”,
r’t’) ,
(7)
depends on the interaction terms. Eq. (7) is represented in terms of
r’, w)G(r”,
r, w) = 6 (r-r’)
,
(8)
where G(r, r’, w) and IZ(r, r’, w) are, respectively, the one-body Green’s function and the selfenergy operator in energy space. w is a parameter which has the dimensions of energy. Eq. (8) is well known as Dyson’s equation. The 270
Volume 47, number 2 Fourier-transformed
CHEMICALPHYSICSLETTERS and energy-dependent
G(r,r’,w)=p
15 April 1977
Green’s function is represented as follows:
(N,O~~(~)~)IN+l,LXN+l,kl~*(~‘)lN,O)(w-(~~+l
+~tN,Ol$*(r’)lN-
l,L’XN-
I,k’l$*(r)lN,
O){w-(E$-l
-E[)tin}-l -Et)-in}-r
,
(9)
where IN, k) and Et represent the kth state of the N-particle system and its energy, respectively. ?‘he factor n is an infinitesimal positive number. Thus eq. (9)ensures that the poles of C(r, r’, o) give the ionization potentials and the electron affinities. In order to obtain the poles from eq. (g), C and Z must be regarded as integral operators, and they are treated in matrix form: {w - [h + Z(w)] )G(w) = 1,
(10)
where 1 is the unit matrix. Therefore, the poles satisfy the following equation, .
Iw-{htX(w))l=O.
(1I)
The ionization potentials are obtained by solving eq. (11). The key points in calculating the ionization potentials are how to select the basis sets and how to lake account of the matrix formalism of Z(w). Z(o) can be expanded by means of the perturbation method on the Creeu’s function. h need not be defined as in eq. (1) but can be chosen as in the one-electron operator of the HF type. In this paper the latter operator is adopted. The use of the diagonalized representation of the number of the particle gives Has H=H’+H’,
Ho = C i
eiai+ai , H’ = ,g
Cj Ikl)aTaj’alak -
c Viiataj , u
(12)
where I denotes occupied orbitals, and fi is an orbital energy. Here the orbital includes the spin (any spin), and the following abbreviation is used: (ij(kl) =Sdrdr’lpi”(r)lpi”(r’)(e211r - r’l)q&)q(r’)
.
The ai* and er are the creation and annihilation operators for electrons in orbital i, and satisfy the following relations: (Gi,ai*)=6ij, H’ is the perturbation,
{ai,aj}=
{af.ai*)=O.
and Z(U) can be expanded in a perturbation
(13) series:
Z(w) = X(‘)(w) + Z “VW) t Z@)(U) t . .. , where Z(‘)(w) is equal to zero if HF orbitals are adopted, Xc2)(,) is listed in ref. [S] , and Zt3)(uj by using the diagrams of fig. 1. Then the equation to be solved is
(14) is formul~~cd
Iw-e-Z(w)l=O.
(15)
where the E is the diagonal matrix of the orbital energies. When C = CCli, this formaiism coincides with an iiF determinant.
271
Volume 47, number 2
CHEMICALPHYSICSLETTERS
15 April 1977
3. Results and discussion
We have calculated the valence and core ionization potentials (the principal lines) of Ne, Mg atoms and Hz0 by using the theory of section 2 and the following methods: (I) The one-center expansion method [IS] based on Slater-type functions is used for the waveftmction of the ground state. The orbital exponents are selected as follows, Ne: ls(12.51,7.88), Zs(3.?5,2.3),2p(3.75,2.3), 3s(l.S), 3p(lS); Mg: ls(15.09,9.52), 2s(4.81,3.03), 2p(5.09,3.21), 3s(1.43,0.91), [email protected],4s(O.91). (2) The second- and third-order perturbation terms Z.“2)(o) and Z ‘(GI) of the self-energy part are formulated by using similar Feynman diagrams to those by Thouless [ 161, which are listed in fig. 1.
rig. I. Second- and third-order diagrams contributing to C(G).
The third-order perturbation term is partitioned as X”)(w) = C(3)(2) + I@(l)
+ Z’S’(O) 1
where Zt3)(2), Zt3)(1), and Z’3’(O) are the 02-, o-dependent, and w-independent terms, respectively. Among these three terms, the largest km is s3)(1). This has been checked up by the numerical calculation of the valence ionization potentials [6,17]. However; the self-energy corrections to the HF approximation are 2-S eV in the valence region, and 20-25 eV in the core region. Therefore, we believe that the most important term was the w*-dependent term [Zc3)(2)J for core ionization potentials, and calculate the terms Z’2’(w) and Z(3)f2) of the self-energypart. It will be confirmed by the calculations of the Is levels of Ne and Mg atoms whether the above approximation is reasonable or not. The calculated results are listed in table 1. Z’2D’(w) implies th at only the dia onal parts of Z”(w) were caIcuIated, and Z(3D)(o) denotes that the diagonal parts of Z(o) were calculated by ZQ2)(o) plus the w2-dependent terms of C(3)(w), and the non-diagonal parts are calculated by the 2(2)(w) method. The calculated results from Zt3D) (0) compare weIf with the experimental data [18] in the core region, but do not give good rest&s in the valence region in comparison with Cederbaum’s calculations [lo] which inchrded more types of diagrams. Therefore, we believe that the above approximation methods are reasonable for the principal line of the core region, but are not appropriate for valence ionizations if resuhs within 0.3 eV of the experimental value are required. 272
CHEMICAL PHYSICS LETTERS
Volume 47, number 2 Table 1 The ionization
potentials
of neon and magnesium
(ev)a)
Ne
HF d2W”) c(%& z@“)(w) exptl.
15 ApriI f977
Mg
IS
2s
2P
Is
2s
2P
3F
894.5 858.6 868.7 871.8 870.7
53.2 47.1 48.3 49.7 48.5
23.1 18.6 19.1 20.2 21.6
1332 1303 1308 1309 1312
100.2 95.6 96.2 96.1 95.2
58.9 54.0 54.6 54.6 55.8
6.8 7-I. 7.1 7-E 7.6
P (%), a) HF denotes the Hartree-Fock approximation_ 2(2R&), and XeD& w 1 represent the caIcuIations including secondorder perturbation terms of G(w), for oniy the diagonal matrix elements of C(*%w), and for the non-diagonal mat&x elements using the w*-dependent term of the third-order perturbation terms of C(w) for the diagonal matrix elements, respectively. ETperimental values are from ref. [IS].
We calculated the ionization potentials of H20 by the above method_ The orbital exponents were selected follows, H20: ls(12.6,7.45), 2s(2.2458), 2p(2.2266), 3s(2.005), 3p(l.838), 3d(l.5), 4s( l-946), 4p(1-9). Table 2 The ionization
potentiais
as
of Hz0 (C,v) (ev)a)
H20
la1
2ai
lb1
Sal
HF .-P)(w) d2~(d *D)(w) exptl.
561.2 531.2 540.5 541.6 539.9
37.0 32.2 32.0 32.0 32.2
18.7 16.7 16.9 17.2 18.4
15.5 12.7 13.1 13.5 14.7
a) Expe~mental
values are from ref. 1191.
13.7 10.8 11.1 11.6 12.6
‘l&e calculated results (tabIe 2) for the 1al and 2al levels from the Z( 3D)(~) method are in good accord with the experimental data [19], but the results for the valence region (except 2al) do not compare well with the results of other calculations [8,14]. This is confirmed by the calculations on Ne and Mg. Thus we believe that the E(3D)(U) method is appropriate for the cakuiation of core ionization potentials.
References fl] L.D. Landau and E-M. Liisbitz, Quantum mechanics (Pergamon Press, London, 1964). [2] W. Meyer, Intern. 3. Quantum Chem. 5s (1971) 341; J. Chem. Phys 58 (1973) 1017; T.H. Dunning, R-M. Pitzer and S. Aung, 3. Chem. Phys. 57 (1972) 5044. [3] D.B. Adams and D.T. Clark, Tbeoret. Chim. Acta 31 (1973) 171. [4] M.H. Wood, Chem. Phys. 5 (1974) 471. [S] J.D. Doll and W.P. Reinhardt, J. Chem. Phys 57 (1972) 1169. [6 1 L.S. Cederbaum, Theoret. Chim. Acta 3 l(1973) 239. (71 L-S. Cederbaum, G. Hohlneicher and W. von Niessen, Chem. Phys. Letters 18 (1973) 503.
273
Volume 47. number 2
CHEMICAL PHYSICS LETTERS
15 April 1977
ffi] L.S. Cederbaum, Mol. Phys. 28 (1974) 479. [Pf W. von Niesson, W-P. Kraemer and L.S. Cederbaum, C&em. Phys. 11 (1975) 385. [IO] LS. Cederbaum and W. von Niessen, Cbem. Phys. Letters 24 (1974) 263. 1111 W. von N&en. G.H.F. Diercksen and L.S. Cederbaum. Chem. Phvs 10 (1975) 345.459. ji2j W. von Niessen; LS. Cederbaum, G.H.F. Diercksen and G. HohIn~iche~,&em. Phyi 11 (1975) 399, f131 L.S. Cederbaum. J. Chem. Fhvs. 62 (1975) 2i60. ii4 j D-P. Chong, F-6. Herring aniD_ &&&l&is, J. Cbem. Phys. 61 (1974) ?8,9.58,356?. 1151 H. Hartman, I, Papula and W. StehI, Theoret. Cbim. Acta 17 (1970) 131; 19 (1970) 155. f16] D.3. Thouies, The quantum mechanics of many-body systems (Academic Fress, New York, 1961). Ii71 L-S. Cederbaum. J. Phys B8 (1975) 290. (181 J.C. SIater, Phys Rev. 98 (1955) 1039. f 191 K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Hedbn, K. Ham&, U. Gehus, T. Bergmark, L.O. Werme, R. MaMe and Y. Baer. ESCA appiied to free molecules (North-Holland, Amsterdam, 1969).
274
CALCULATION OF VFlTICAL
CHEMICAL PHYSICS LETTERS
15 April 1977
IONIZATION POTENTIALS
USING THE ONBBOL 1(GREEN’S FUNCTION: NE, Mg, and H,O H. YAMAKAWA, T. AOYAMA Department of Chemistry, Faculty of Science, the Universt.:) of Tokyo, Hongo. Bunkyo-ku. Tokyo, Japan
and H. ICHIKAWA Division of ChemicalPhysics, Hoshi Collqe cf Pharmacy, Z-4 Ebara, Shinagawa-ku, Tokyo, Japan
Received 6 September 1976 Revised manuscript received 5 November 1976
The valence and core ionization potentials of Ne, Mg, and Hz0 have been calculated by using the one-body Green’s function. The self-energy part WBStaken to include the second order and the w2dcpcndent terms, third order ita the per-
turbation.
1. Introduction Recent development of photoelectron spectroscopy has made it an indispensable tool for establishing and clarifying orbital schemes of organic and inorganic molecules. The Hartree-Fock approximation (HF) and Koopm.ns’ theorem are usually applied to the analysis of the photoelectron spectra. However, it is well known that calculations based on the HF wavefunction and on Koopmans’ theorem give values iarger than those of experiment. ‘?ne difference between the calculated and experimental values (Koopmans’ defects) turn out to be several eV in the case of valence ionization potentials and over 10 eV in the case of core ionization potentials. A theoretical method must therefore remove the Koopman’s defect. Shtce the sudden approximation [l] may hold in photoelectron spectroscopy’, the ionization potentials (A&,, j may be expressed as AEon = E(N- 1, n) - E(N, 0) , where E(N, 0) is the energy of the ground state of the initial state of the N-partick system, and Et/V-I, n) is the energy of the nth state of the final state [(/V-l)-particle system]. Several approximate methods [2-6] which evaluate the energy difference between these two states have beer presented. Two methods are based on the application of many-body perturbation theory (5,G]. This utilises the property of the poles of the one-body Green’s function being the ionization potentials and the electron afficiti?s Some formalisms which consist of different approximations for the self-energy part Z(w) have been presented t!, Doll and Reinhardt [5! and Cederbaum and co-workers [f&12]. Their results compare weil with the erperimenmi values of rhe valence ionization potentials, but so far no emphasis has been given to core icnizntion potentials. ft is known that the single-particle-excitation approxtmation does not apply to the case of core-ionir*jtion, since intcnsc satellite lines are observed together with the principal line [ 131. The satellite lines are not calculated here by usiris
CHEMICAL
Volume 47. number 2
PHYSICS LETTERS
15 April 1977
the theory of the one-body Green’s function. We believe however that the principal lines may be calculated in this way. The principal lines have here been caiculated by using another theoretical method within the single-particleexcitation approximation [ 143. We have calculated the valence and core ionization potentials (the principal lines) of Ne, Mg, HzO, and compare them with the experimental data.
2. Theory The hamiltonian H of the molecular system is, H=
~h,+r~-“21~i-~+-‘,
hi = -(A2/2m)ViZ + Ce2 4
Ir, - ‘;-I-l ,
(1)
where i, j, and o[ represent the electrons and nuclei respectively. The behavior of the electrons in the molecular system is deduced by solving the following Schrbdinger equation or eigenvalue problem: H#
=iliagi/at,
Hq = Ep,
(2)
where 9 is usually represented by a Slater determinant to account for the Pauli principle. The practical method in which the above concept may be satisfied, the Hartree-Fock method, however, has the drawback that the interactions between
electrons
are not appropriately
included.
As is well known,
it is important
that electron
correla-
tion should be included in a transparent form. In second quantized form, the physical quantities are represented by the creation 9*(r) and annihilation G(r) field operators. The hamiltonian is H=J*JI*(r)h(r)rCl(r)r)
+ ~~~‘JI*(r)lL*(r’)(e2/1,-
r’l)$(r’)lC
(r) ,
(3)
where 9 *, 3/ satisfy the following relationships: U(r), 3/?‘)3
U(r), $@‘)I = C@*(r), $*@‘)I =‘O ,
= 6 (r-r’),
(4)
where { } is the anticommutator, {A, B} = AB + BA. The hatniltonian of eq. (3) does not contain an explicit number of particles. Therefore, such an expression for the hamiltonian is more general than that of eq. (I), since a change in the number of particles is included naturally. Thus the direct calculation of a physical quantity such as an ionization potential, which arises from a change in the number of particles, is possible by the hamiltonian expressed by eq. (3). Here we use the Heisenberg representation for convenience: ifi a+(r, t)lat = ]9(r, t),rrl ,
(5)
where [ ] is the commutator, [A, B] = AB - BA. The ground state of the IV-particle system is represented by ]lv, O>,and the one-body Green’s function is defined by the equation: G(rt,r’f’)=i-‘UV,OIT
[~,!~(r,t).$*(r’. r’)] IN, 01,
(6)
where T is Wick’s time-ordering operator. Eqs. (5) and (6) lead to the equation of motion for the Green’s function {i a/at -h(r))
G(rt, r’t’) = 6 (t - t’)6 (r - r’) +~dt”dr”Z(rt,
where the operator Z(rt, r’t’), the selfenergy, w by the Fourier transform over (t - t’) {o - h(r)) G(r, r’, w) -&r”Z(r,
r”t”)G(r”t”,
r’t’) ,
(7)
depends on the interaction terms. Eq. (7) is represented in terms of
r’, w)G(r”,
r, w) = 6 (r-r’)
,
(8)
where G(r, r’, w) and IZ(r, r’, w) are, respectively, the one-body Green’s function and the selfenergy operator in energy space. w is a parameter which has the dimensions of energy. Eq. (8) is well known as Dyson’s equation. The 270
Volume 47, number 2 Fourier-transformed
CHEMICALPHYSICSLETTERS and energy-dependent
G(r,r’,w)=p
15 April 1977
Green’s function is represented as follows:
(N,O~~(~)~)IN+l,LXN+l,kl~*(~‘)lN,O)(w-(~~+l
+~tN,Ol$*(r’)lN-
l,L’XN-
I,k’l$*(r)lN,
O){w-(E$-l
-E[)tin}-l -Et)-in}-r
,
(9)
where IN, k) and Et represent the kth state of the N-particle system and its energy, respectively. ?‘he factor n is an infinitesimal positive number. Thus eq. (9)ensures that the poles of C(r, r’, o) give the ionization potentials and the electron affinities. In order to obtain the poles from eq. (g), C and Z must be regarded as integral operators, and they are treated in matrix form: {w - [h + Z(w)] )G(w) = 1,
(10)
where 1 is the unit matrix. Therefore, the poles satisfy the following equation, .
Iw-{htX(w))l=O.
(1I)
The ionization potentials are obtained by solving eq. (11). The key points in calculating the ionization potentials are how to select the basis sets and how to lake account of the matrix formalism of Z(w). Z(o) can be expanded by means of the perturbation method on the Creeu’s function. h need not be defined as in eq. (1) but can be chosen as in the one-electron operator of the HF type. In this paper the latter operator is adopted. The use of the diagonalized representation of the number of the particle gives Has H=H’+H’,
Ho = C i
eiai+ai , H’ = ,g
Cj Ikl)aTaj’alak -
c Viiataj , u
(12)
where I denotes occupied orbitals, and fi is an orbital energy. Here the orbital includes the spin (any spin), and the following abbreviation is used: (ij(kl) =Sdrdr’lpi”(r)lpi”(r’)(e211r - r’l)q&)q(r’)
.
The ai* and er are the creation and annihilation operators for electrons in orbital i, and satisfy the following relations: (Gi,ai*)=6ij, H’ is the perturbation,
{ai,aj}=
{af.ai*)=O.
and Z(U) can be expanded in a perturbation
(13) series:
Z(w) = X(‘)(w) + Z “VW) t Z@)(U) t . .. , where Z(‘)(w) is equal to zero if HF orbitals are adopted, Xc2)(,) is listed in ref. [S] , and Zt3)(uj by using the diagrams of fig. 1. Then the equation to be solved is
(14) is formul~~cd
Iw-e-Z(w)l=O.
(15)
where the E is the diagonal matrix of the orbital energies. When C = CCli, this formaiism coincides with an iiF determinant.
271
Volume 47, number 2
CHEMICALPHYSICSLETTERS
15 April 1977
3. Results and discussion
We have calculated the valence and core ionization potentials (the principal lines) of Ne, Mg atoms and Hz0 by using the theory of section 2 and the following methods: (I) The one-center expansion method [IS] based on Slater-type functions is used for the waveftmction of the ground state. The orbital exponents are selected as follows, Ne: ls(12.51,7.88), Zs(3.?5,2.3),2p(3.75,2.3), 3s(l.S), 3p(lS); Mg: ls(15.09,9.52), 2s(4.81,3.03), 2p(5.09,3.21), 3s(1.43,0.91), [email protected],4s(O.91). (2) The second- and third-order perturbation terms Z.“2)(o) and Z ‘(GI) of the self-energy part are formulated by using similar Feynman diagrams to those by Thouless [ 161, which are listed in fig. 1.
rig. I. Second- and third-order diagrams contributing to C(G).
The third-order perturbation term is partitioned as X”)(w) = C(3)(2) + I@(l)
+ Z’S’(O) 1
where Zt3)(2), Zt3)(1), and Z’3’(O) are the 02-, o-dependent, and w-independent terms, respectively. Among these three terms, the largest km is s3)(1). This has been checked up by the numerical calculation of the valence ionization potentials [6,17]. However; the self-energy corrections to the HF approximation are 2-S eV in the valence region, and 20-25 eV in the core region. Therefore, we believe that the most important term was the w*-dependent term [Zc3)(2)J for core ionization potentials, and calculate the terms Z’2’(w) and Z(3)f2) of the self-energypart. It will be confirmed by the calculations of the Is levels of Ne and Mg atoms whether the above approximation is reasonable or not. The calculated results are listed in table 1. Z’2D’(w) implies th at only the dia onal parts of Z”(w) were caIcuIated, and Z(3D)(o) denotes that the diagonal parts of Z(o) were calculated by ZQ2)(o) plus the w2-dependent terms of C(3)(w), and the non-diagonal parts are calculated by the 2(2)(w) method. The calculated results from Zt3D) (0) compare weIf with the experimental data [18] in the core region, but do not give good rest&s in the valence region in comparison with Cederbaum’s calculations [lo] which inchrded more types of diagrams. Therefore, we believe that the above approximation methods are reasonable for the principal line of the core region, but are not appropriate for valence ionizations if resuhs within 0.3 eV of the experimental value are required. 272
CHEMICAL PHYSICS LETTERS
Volume 47, number 2 Table 1 The ionization
potentials
of neon and magnesium
(ev)a)
Ne
HF d2W”) c(%& z@“)(w) exptl.
15 ApriI f977
Mg
IS
2s
2P
Is
2s
2P
3F
894.5 858.6 868.7 871.8 870.7
53.2 47.1 48.3 49.7 48.5
23.1 18.6 19.1 20.2 21.6
1332 1303 1308 1309 1312
100.2 95.6 96.2 96.1 95.2
58.9 54.0 54.6 54.6 55.8
6.8 7-I. 7.1 7-E 7.6
P (%), a) HF denotes the Hartree-Fock approximation_ 2(2R&), and XeD& w 1 represent the caIcuIations including secondorder perturbation terms of G(w), for oniy the diagonal matrix elements of C(*%w), and for the non-diagonal mat&x elements using the w*-dependent term of the third-order perturbation terms of C(w) for the diagonal matrix elements, respectively. ETperimental values are from ref. [IS].
We calculated the ionization potentials of H20 by the above method_ The orbital exponents were selected follows, H20: ls(12.6,7.45), 2s(2.2458), 2p(2.2266), 3s(2.005), 3p(l.838), 3d(l.5), 4s( l-946), 4p(1-9). Table 2 The ionization
potentiais
as
of Hz0 (C,v) (ev)a)
H20
la1
2ai
lb1
Sal
HF .-P)(w) d2~(d *D)(w) exptl.
561.2 531.2 540.5 541.6 539.9
37.0 32.2 32.0 32.0 32.2
18.7 16.7 16.9 17.2 18.4
15.5 12.7 13.1 13.5 14.7
a) Expe~mental
values are from ref. 1191.
13.7 10.8 11.1 11.6 12.6
‘l&e calculated results (tabIe 2) for the 1al and 2al levels from the Z( 3D)(~) method are in good accord with the experimental data [19], but the results for the valence region (except 2al) do not compare well with the results of other calculations [8,14]. This is confirmed by the calculations on Ne and Mg. Thus we believe that the E(3D)(U) method is appropriate for the cakuiation of core ionization potentials.
References fl] L.D. Landau and E-M. Liisbitz, Quantum mechanics (Pergamon Press, London, 1964). [2] W. Meyer, Intern. 3. Quantum Chem. 5s (1971) 341; J. Chem. Phys 58 (1973) 1017; T.H. Dunning, R-M. Pitzer and S. Aung, 3. Chem. Phys. 57 (1972) 5044. [3] D.B. Adams and D.T. Clark, Tbeoret. Chim. Acta 31 (1973) 171. [4] M.H. Wood, Chem. Phys. 5 (1974) 471. [S] J.D. Doll and W.P. Reinhardt, J. Chem. Phys 57 (1972) 1169. [6 1 L.S. Cederbaum, Theoret. Chim. Acta 3 l(1973) 239. (71 L-S. Cederbaum, G. Hohlneicher and W. von Niessen, Chem. Phys. Letters 18 (1973) 503.
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CHEMICAL PHYSICS LETTERS
15 April 1977
ffi] L.S. Cederbaum, Mol. Phys. 28 (1974) 479. [Pf W. von Niesson, W-P. Kraemer and L.S. Cederbaum, C&em. Phys. 11 (1975) 385. [IO] LS. Cederbaum and W. von Niessen, Cbem. Phys. Letters 24 (1974) 263. 1111 W. von N&en. G.H.F. Diercksen and L.S. Cederbaum. Chem. Phvs 10 (1975) 345.459. ji2j W. von Niessen; LS. Cederbaum, G.H.F. Diercksen and G. HohIn~iche~,&em. Phyi 11 (1975) 399, f131 L.S. Cederbaum. J. Chem. Fhvs. 62 (1975) 2i60. ii4 j D-P. Chong, F-6. Herring aniD_ &&&l&is, J. Cbem. Phys. 61 (1974) ?8,9.58,356?. 1151 H. Hartman, I, Papula and W. StehI, Theoret. Cbim. Acta 17 (1970) 131; 19 (1970) 155. f16] D.3. Thouies, The quantum mechanics of many-body systems (Academic Fress, New York, 1961). Ii71 L-S. Cederbaum. J. Phys B8 (1975) 290. (181 J.C. SIater, Phys Rev. 98 (1955) 1039. f 191 K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Hedbn, K. Ham&, U. Gehus, T. Bergmark, L.O. Werme, R. MaMe and Y. Baer. ESCA appiied to free molecules (North-Holland, Amsterdam, 1969).
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