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Direct calculation of energy differences by a common unitary transformation of two model states, with application to ionization potentials
Volume 66, number 1
CHEMICAL
15 September 1979
PHYSICS LE’ITERS
DIRECT CALCULATION OF ENERGY DIFFERENCES BY A COMMON UNITARY OF TWO MODEL STATES, WITH APPLICATION TO IONIZATION POTENTLALS
TRANSFORMATION
H. REITZ and W_ KUTZELNICG Lehrsttdd $icr T.%eoretisclte Clremie der Rtdtr- ffnii ersitut Bochunt. D-4630 Bochum. Federal Republic of Germaqv
Received 1 June 1979
Tote approtiate wavefunctions Qo and *’ for tv,o states can be transformed by a common unitary operator (I = exp(o), O+ = - (I, to the corresponding esact states 90 and w’_ The energy correction to second order in o is evaluated with (I chosen such that it makes the ener_p)’difference stationary. The contributions of various types of excitation operato:s conmined 51 D are analyzed. Contriiutions that are identical for the two states came1 automatically. Numerical results for Hz0 and Ne indicate that the Koopmans correction is to a large extent recovered and given in terms of physically inter-
I_ Introduction Standard quantum chemical calcuIations yield total energies Ek of eIectronic states of atoms or molecules. Physically relevant are only energy differences Mx- = E, - EI, that are then obtained as small differences between large numbers. This is why one has tried for a long while to find methods that Iead ‘*directly” to such energy differences_ For the special case of ionization potentials the most popular “direct” methods are (1) application
of Rs perturbation
theory to s
[l--31. (2) the so-called
EOM (equations-of-motion
14-61, (3) determination of the poles of one-particle functions [7--12), (4) methods
that include correlation
cluster expansion Relations
between
by several authors
method) Green-s
effects by a
[ 13,14]_
these approaches
have been discussed
[ 15,16]_
We present here a new method,
that is conceptually more closely reIated to the standard approaches for total energies_ It is basicaIIy variational, but it takes care of explicit cancellations of correIation contributions to the ground state of the neutral moIecuIe and the considered state of the ion_
A detailed account of the theory shaI.l be given elsewhere_ Here we limit ourselves to a short description of the formalism and the discussion of some results. Our formalism can also be used for spectral transition energies.
2. Unitary transformation
of model states
We are interested in the eigenstates Xl!, and eigenvalues Ek of a hamiltonian
H. We define H in Fock
space in order to habe a common hamiltonian for the states of the ion and those of the neutral moIecuIe_ We assume that approximate wavefunctions (model wavefunctions) cbk are known for those states that we are interested ir.. and that these model wavefunctions are orthonormai_ iGx- I :I,,> =
6 k[_
(1)
The [I+ are, so far, Lzmpletely arbitrary otherwise_ There exists at least one unitary transformation * U that transforms the ‘4 to the exacf ‘I%i and a unitary transformation can always be written in the form ’ U is not uniquely determined by the requirement U@k = *k for two states, one can hence impose additional restrictions on 19 to tie it unique. 111
u=e=.
15 September 1979
Cfff3fICAL YftYfSCS LE-i-f-ERS
Volume 66. number 1
u+= -u_
(3
KnowJedge of tJIe antiflermitcm operator u impJies of the +& and the &i, and of the energy differences
one expands u in terms of a full basis in operator as foJJows u=ut
space
tf7-J +___a,,
knowkdge
x-,
= C*r;, Jt-“He”[
aI@ - <+~Je-“HeuJ~J~l)_
(3
If we expand e-“He“ according to the ttausdorff form& f 17 1, EL, becomes an expansion in powers of u with the zeroth-order term 3fzz = <‘ID,IHI ‘I’_$ - <‘l~~lHI‘I’~~*
(4)
i-e_ the energy difference corresponding to the model functions. We nest assume titat 9, is tile model ground state of tJle neutral nxolecule nnd 81~’the considered model state of tile ion (for an excited state of the neutral moJecuJe the t-ormaJism is the same) which are rehted througJt a model ionization (or excitation) operator R* that satisfies (5)
fg
= _fP? xc7
etc_ In order to caJcuIate 4E from (7) one needs additionaJJy a procedure to construct an appropriate u, or equitientiy the coefficients f in (8). After con.sLiering different possibilities we found that a very zffectivvc way is to dloose thefsucJ1 tllat tJiey make LE stationary_ We further simplified the formalism bb considering only crl and cr2 and by truncating t1.e Zfausdorff expansion after tile term quadratic in u. The expression to be made stationary is then (9)
(6) it is recummended that ,O,* has a simpJe structure in the second quantization formafism, e-g_ that for ionization it is jus rk snnihihrion operator t-or one orbitai (occupied in the Sizer determinant model function +o)One then sets for the ionization (orescitation) -V; = <+OJOe-uI~eo~’
energy
- e-PHe*QQ*~cPO>
with
A
R
B
RS
wo~[[%W, RI I,
=
=
f
@J I+,),
~(c4@[[Q~[[H,RI .Sll, Q%I@
Wo! [ [Q, [[If_ SI , R] 1,
,o,+l+,)),
(JO)
where R- stands for any one- or two-particle basis operator such as (alaq - a”0 p p ) or aFazaqap - a;aGaflr and f, is the corresponding expansion coefficient_ Minimization of (9) with respect to the fk yields
AR *
+ :<+,r
[ f-O-_[[H, QJ , 01 J . 2’1
i+t)> f __. _
(7)
Expression (7) for rfle ionization energy is f&y Liealgebraic_ i-e_ onIy [inked terms contribute and aJJ contributions that are the j-3me for Et, and E’ cancel ;lutomaricaliy_ TO take full advantage of tile Lie-algebraic structure
+&_
=0,
(II)
whicJl is a linear system, the solution of which is not too difficult_ For the fs that stisfy (I I) one gets
gfRAR
l
RcsfRfsBRs .
and together with (9)
= 0
(12)
Volume 66, number 1
The two simplifying assumptions (limitation to or and 02 and truncation of the Hausdorff expansion after U(o’)) are on the same level as the best current methods for total energies of individual states (see eg. ref. [IS] ) where they have turned out to be rather satisfactory. So we do not espect significant errors due to the approximations. Somewhat more critical is the determination of thefR from a stationarity principle on the energy difference_ A theoretical justification wiII be given in the fuli account of the method_ We point out that our scheme is practictiIIy equivalent to separate CEPA-0 calculations (see ref_ [IS] ) for the two states and subsequent formation of the difference_ It is also equivalent to third order perturbation theory with a renormalization correction_
3. CIassification
of the basis operators
We consider closed she11 states for the neutral molecules (or atoms) and use the model wavefunction +0 * as a single SIater determinant with rhe MO’s Us occupied_ The model ionization opemtor is ,O,+ =
ai_
i.e. it annihilates the occupied MO pii- The operator basis {RI consists then on spin orbital level of operntors of different types, such ;ts (the subscriptsj, A_(+ [) are used for occupied unoccupied orbit&)
orbit&
other than 9;: G. b for
(1) external single substitutions if Q,Qj
-
ai Q,;
single substitutions from Us QiQr - Q:QQ; (3) internal singIe substitutions i(3)
Qi Qi
-
Qifaiir
(4) e.xtermd double substitutions Q,fabt’lx-Qj (5)
-
Q;Q&Q,
;
double substitutions Q;Q;QkQi
-
involving qi
-
(i,’ (in which ef is empty)_ The action of 7 on +, is the same as of 1 on +o. It is therefore recommended to “orthogomdize” 1 to 7 and to replace I by (I’) CompIementary Q;Q,Q&
single substitution
= Q;Qi - Q;Q;QrQi_
Now I’ acts only on G’ and the only operators acting on both @j, and @’ are those of type 4_ These. however, give (to second order in a) oniy unlinked terms, that do not contribute to A,!? and that can hence be ignored_ If $-, describes a closed-shell ground state_ we use a restricted Hartree-pock function with doubly occupied orbitais. It is then recommended to rep&e the substitution operators just given on spin orbital Ievcl by spin-adapted substitution operators on orbital level_ Instead of 7 types of opemtors one now gets IS different types, 5 of which correspond to external double substitutions and do not contribute to X FuII expressions of the remaining 13 operators will be given elsewhere. Here we limit ourselves to a pictorial classification. We assume that the orbital &- with &spin is ionized (a bar always designates P-spin). There are 3 types that act only on +,r (a) single substitutions from Zi: (b)doubIe substitutions from pi and pi to FQ and GQ; (c) double substitutions from tir-and Zr-to cs_ Z6 and GoTgbSeven types act only on +‘r (d)intemaI single substitutions (from $ to Zi): (e) conditional singie substitutions from “i (only if Zr is empty); (f) “relaxation” (substitutions from q,. to qb or Gi to Gb if hi iS empty); (g) “spin-polarization”; (11). (i). (j) semi-internal coupIing schemes_ fhree
double substitutions,
types act simultaneously
different
on +,, and ~1)‘:
Q;Q;QbQ,;
(6) semi-internal double substitutions t-r+-+ Q0Qi QkQ,- - Q,- Q,@& ; (7) conditionai single substitutions QzQTQpj
15 September 1979
CHEMICAL PHYSICS LETTERS
Q;Q;Q~=
_
The operators of type $,5 and 7 act only on $, (in which pi is occupied), those of type 3 and 6 only on
(k), @I)_ (m) doubie substitutions coupling schemes_ These 13 types of matrix elements tion the respective ed by a computer
involving ai_ different
of basis opemtors lead to 169 types BRs_ To avoid errors in their evahraformulas for the BRs were generatprogram. 113
Volume 66. number I
5_ Numerical resuks
4.Solution of the linear system The solution of the Liner system (I 1) is possible in a brute-force way- It is, however, better to proceed iterativeIy, namely to start with
04) and to obt& (01
BRRfR
l&*1)
the@*f) =
_-cp
R -
from c 8RS@’ S(*R)
until se!fconsistency_ One advantage of this scheme is tkt the matrix BRs need not be evJIuated as such (which would require a fuII twoeIectron integral trzcnsformation), but onIy tile VeCtOr ~S(_R$~fS which requires just one pzss through the i&egg! t;lpe_ The program used is based on the integral package described in ref. [I91 and the SCF part of the CEPA proo,clm described in ref_ [ZOl_
7s3P 9sSpld 9dp2d experiment [lZl
CEPA. IIsQ4dif
15 September 1979
CHEMICAL PHYSICS LETl-ERS
[Xl
Grcen’sfunction.s, 1ls7p7d [??I-
We have applied the method to several molecules and atoms_ In the present note we report the results for two representative systems, nsmefy H,O and Ne_ Tribe I ilhtstrates the dependence of the results on the basis of gaussian lobes- One sees that a sufficient number of poknization functions is necessary for the resuk to converge tolwrds the experimental vaIues_ We ~ISO compare with results from the literature and from experiment. In table 2 the total ionization potentials are decomposed into contributions of the model level (Koopmans energies) and various corrections_ The corresponding resuIts for Ne are given in table 3. One sees that the main corrections to the Koopmans values of ionization potent& of the vaience electrons come from the operators that describe “relaxation” (type (e) and (0) i-e_ that allow for the “Lremaining” orbit& in the ions to relax, and from the external double substitutions that involve yi (types (b), (c), (k), (n), (i-u))_ These types of contributions have the same order of magnitude, but opposite sign, so their contributions cancel to a large extent 1211 and the total correction is relatively small. The situ3tion is somewhat
lb2
lb,
3Zl
0.41937
O-49631
--0.67859
O-46154 O-46763 0.4698 0_45854 0.4730
0.54314 051936 0.545 f 0.004 OJ3950 0.5457
0.69999 0.70147 0.688 f 0.008 0.69286 0.6975
Ia1
19.79640 3)
18.821 19.83077
a) Basis 95Sp (uncontmcted) + J steep p-function. Table 2 Contributions to the ionization potenti& for water, basis: 9sSpZd (9Sp f up for lal)
IhopmsnsIp
=k..tfon
(types(e), (0)
spin po&rization &) externzddouble substitutions (b. c, k. P, m) semi-intennl double substitutions (lx. i, j) result
114
in au
-0.08895
-0.08099
0.71059 -0.05888
-0.01020 0.07346 -0.0095s G-46736
-0.00844 0.07451 -0.01368 054936
-0.00896 0.07447 -0.01576 010147
050290
0.57797
3055554 -0.77621 -0.02666 0.04166 0.00206 19.79640
Volume 66, number 1
CHEMICAL
TabIe 3 IonWtion potentktls for neon, basis: 9s5p2dlf (in 7-P Koopmans
fP
r&.kation spin pofarizztion external double substitutions semi-internal double substitutions result experiment 22.51
0.84465 -0.10687 -0.01081 0.08147 -0.01432 O-79413 0.7938
PHYSICS LJZTTERS
15 September 1979
References au>
2s
1.92392 -0.10668 -0.00942 0.04858 -0.07396 1_78245 1.7813
different for ionization from the core (Is)_ Here all corrections other than the relaxation are very sm3Il. such that relaxation dominates by far_ Sin&w observations bttve been made by other authors [22,23], based on different methods_ The comparison of the various contributions from different sources is not always easy. since these contributions cx! be defined differently. We have used the contributions $ fRAR that according to (is), sum up to the total Koopmans defect. The agreement of our results with experiment is of about the same quality as that of competing schemes. The remaining error is of the order of IO--2070 of the Koopmans error. It sItouId be mentioned that the metbod in the present form fails for certain ionizations from orbit& tbsr are between valence zrndcore orbit& (like 2a1 in fI20). The one-particle excitation is then probably n poor model excitation operator. The fact that most of the ionic states that are invoked in our calculations are strictly not bound states does otherwise not seem to cause problems-
Acknowledgement
El] J-P_ Xfialrieu,J. Chem_ Phyr 47 (1967) 4555. [2] D-P. Chong, F-G. Herring and D_ Mc\\rims, J_ Chem. Phys. 61 (1974) 78.. [3] V. KvamiCka and I. IiubnE, J. Chem. Phys. 60 (1974) 4483.
D.J. Rowe, Rev. Mod. Phys. 40 (1966) 153. IS] J. Simonsand WD. Smith, J_ #em. Phys. 58 (1973) 4899. 161 &f-F- Herman, D-L- Yeaser, K.F. Freed and V. McKay, Chem. Phys. Letters 46 (1977) i_ [7] J-D. Dolf and W.P. Reinhardt, J. Chem. Phys. 57 (1972) 1169.. JS] F. Ecker and G. Wohlncichcr, Theoret. Chim. Acta 25 _ (1972) 289. [V] L-S- Cederbaum, Tfxoret. Chim. Acta 31(1973) 239. [IO] L-S. Cederbaum and ‘w, Domeke. Advm. Chem. PhFs_ 36 (1977) 205. fill B-T_ Pickup and 0. Goscinski, Xfol. Phys. 26 (1973) 1013. [12] G.D.Pu~~andY.bhm,J.Chem. Phss. 60 (1974) 406% 1131 1. PaIdus. J_ &eli, 3L Saute zxd A_ L&orgue. Phys_ Rev_ Al7 (1978) SOS. 114j A.. Izfukhopzdhyay, TX. Moitm and D. 5f ukherjee, J_ Pbys_ B 12 (1979) 1. [l-5] SI.F_ Herman, D-L. Yeager and K.F. Freed, Chem. Ph>s_ 29 (1978) 77. (161 G-D. Purvisand 1’. dbrn, J. Chem. Phys. 65 (1976) 917. 1171 F. fhusdorfi: Ber- Verb- Konigt Sachs. Ges. Wi,s_ -sI~tb-Pbys_ Xl_ Leipzjg 58 (1906) 19. [18] %‘. Kutzeln&. --. in: i\Iodern theoretic& cbemistrv_VoL 3. ed. H-F_ Schaefer III (Plenum Pra% New York; i977) ’ p- 129. il9] R. Ahfrichs. Theoret. Chim. Actn 33 (1974) 157. PO1 R. Ahlrichs. H_ Lischb, V_ Staemmfer and IV- KutzeJp~, J. Chem. Phys 62 (1975) E25. Ial R-S. MulEken. J- Chim. Phys. 46 (1949) 497,675 f22] W- Meyer, Intern. J. Quantum Chem. S5 (1971) 341. P.S. Bztgus,Phys- Rev. 139 (1965) A619. g; W. van Niessen. G-H-F_Dierc~ksenand L.S. Cederbarn, J. Chem. Phys_ 67 (1977) 4124. 1251 C-E. Moore, Atomic Energy Levels, NatI. Bur. Std. CucuIm 467, Vol. 1 (1949). [4]
Tbis work was sponsored by Dents&e Forschungsgemeinscbsfr_ We thank Dr_ V. Staemmler and Dr. H. KolImsr for useful comments and suggestions.
115
CHEMICAL
15 September 1979
PHYSICS LE’ITERS
DIRECT CALCULATION OF ENERGY DIFFERENCES BY A COMMON UNITARY OF TWO MODEL STATES, WITH APPLICATION TO IONIZATION POTENTLALS
TRANSFORMATION
H. REITZ and W_ KUTZELNICG Lehrsttdd $icr T.%eoretisclte Clremie der Rtdtr- ffnii ersitut Bochunt. D-4630 Bochum. Federal Republic of Germaqv
Received 1 June 1979
Tote approtiate wavefunctions Qo and *’ for tv,o states can be transformed by a common unitary operator (I = exp(o), O+ = - (I, to the corresponding esact states 90 and w’_ The energy correction to second order in o is evaluated with (I chosen such that it makes the ener_p)’difference stationary. The contributions of various types of excitation operato:s conmined 51 D are analyzed. Contriiutions that are identical for the two states came1 automatically. Numerical results for Hz0 and Ne indicate that the Koopmans correction is to a large extent recovered and given in terms of physically inter-
I_ Introduction Standard quantum chemical calcuIations yield total energies Ek of eIectronic states of atoms or molecules. Physically relevant are only energy differences Mx- = E, - EI, that are then obtained as small differences between large numbers. This is why one has tried for a long while to find methods that Iead ‘*directly” to such energy differences_ For the special case of ionization potentials the most popular “direct” methods are (1) application
of Rs perturbation
theory to s
[l--31. (2) the so-called
EOM (equations-of-motion
14-61, (3) determination of the poles of one-particle functions [7--12), (4) methods
that include correlation
cluster expansion Relations
between
by several authors
method) Green-s
effects by a
[ 13,14]_
these approaches
have been discussed
[ 15,16]_
We present here a new method,
that is conceptually more closely reIated to the standard approaches for total energies_ It is basicaIIy variational, but it takes care of explicit cancellations of correIation contributions to the ground state of the neutral moIecuIe and the considered state of the ion_
A detailed account of the theory shaI.l be given elsewhere_ Here we limit ourselves to a short description of the formalism and the discussion of some results. Our formalism can also be used for spectral transition energies.
2. Unitary transformation
of model states
We are interested in the eigenstates Xl!, and eigenvalues Ek of a hamiltonian
H. We define H in Fock
space in order to habe a common hamiltonian for the states of the ion and those of the neutral moIecuIe_ We assume that approximate wavefunctions (model wavefunctions) cbk are known for those states that we are interested ir.. and that these model wavefunctions are orthonormai_ iGx- I :I,,> =
6 k[_
(1)
The [I+ are, so far, Lzmpletely arbitrary otherwise_ There exists at least one unitary transformation * U that transforms the ‘4 to the exacf ‘I%i and a unitary transformation can always be written in the form ’ U is not uniquely determined by the requirement U@k = *k for two states, one can hence impose additional restrictions on 19 to tie it unique. 111
u=e=.
15 September 1979
Cfff3fICAL YftYfSCS LE-i-f-ERS
Volume 66. number 1
u+= -u_
(3
KnowJedge of tJIe antiflermitcm operator u impJies of the +& and the &i, and of the energy differences
one expands u in terms of a full basis in operator as foJJows u=ut
space
tf7-J +___a,,
knowkdge
x-,
= C*r;, Jt-“He”[
aI@ - <+~Je-“HeuJ~J~l)_
(3
If we expand e-“He“ according to the ttausdorff form& f 17 1, EL, becomes an expansion in powers of u with the zeroth-order term 3fzz = <‘ID,IHI ‘I’_$ - <‘l~~lHI‘I’~~*
(4)
i-e_ the energy difference corresponding to the model functions. We nest assume titat 9, is tile model ground state of tJle neutral nxolecule nnd 81~’the considered model state of tile ion (for an excited state of the neutral moJecuJe the t-ormaJism is the same) which are rehted througJt a model ionization (or excitation) operator R* that satisfies (5)
fg
= _fP? xc7
etc_ In order to caJcuIate 4E from (7) one needs additionaJJy a procedure to construct an appropriate u, or equitientiy the coefficients f in (8). After con.sLiering different possibilities we found that a very zffectivvc way is to dloose thefsucJ1 tllat tJiey make LE stationary_ We further simplified the formalism bb considering only crl and cr2 and by truncating t1.e Zfausdorff expansion after tile term quadratic in u. The expression to be made stationary is then (9)
(6) it is recummended that ,O,* has a simpJe structure in the second quantization formafism, e-g_ that for ionization it is jus rk snnihihrion operator t-or one orbitai (occupied in the Sizer determinant model function +o)One then sets for the ionization (orescitation) -V; = <+OJOe-uI~eo~’
energy
- e-PHe*QQ*~cPO>
with
A
R
B
RS
wo~[[%W, RI I,
=
=
f
@J I+,),
~(c4@[[Q~[[H,RI .Sll, Q%I@
Wo! [ [Q, [[If_ SI , R] 1,
,o,+l+,)),
(JO)
where R- stands for any one- or two-particle basis operator such as (alaq - a”0 p p ) or aFazaqap - a;aGaflr and f, is the corresponding expansion coefficient_ Minimization of (9) with respect to the fk yields
AR *
+ :<+,r
[ f-O-_[[H, QJ , 01 J . 2’1
i+t)> f __. _
(7)
Expression (7) for rfle ionization energy is f&y Liealgebraic_ i-e_ onIy [inked terms contribute and aJJ contributions that are the j-3me for Et, and E’ cancel ;lutomaricaliy_ TO take full advantage of tile Lie-algebraic structure
+&_
=0,
(II)
whicJl is a linear system, the solution of which is not too difficult_ For the fs that stisfy (I I) one gets
gfRAR
l
RcsfRfsBRs .
and together with (9)
= 0
(12)
Volume 66, number 1
The two simplifying assumptions (limitation to or and 02 and truncation of the Hausdorff expansion after U(o’)) are on the same level as the best current methods for total energies of individual states (see eg. ref. [IS] ) where they have turned out to be rather satisfactory. So we do not espect significant errors due to the approximations. Somewhat more critical is the determination of thefR from a stationarity principle on the energy difference_ A theoretical justification wiII be given in the fuli account of the method_ We point out that our scheme is practictiIIy equivalent to separate CEPA-0 calculations (see ref_ [IS] ) for the two states and subsequent formation of the difference_ It is also equivalent to third order perturbation theory with a renormalization correction_
3. CIassification
of the basis operators
We consider closed she11 states for the neutral molecules (or atoms) and use the model wavefunction +0 * as a single SIater determinant with rhe MO’s Us occupied_ The model ionization opemtor is ,O,+ =
ai_
i.e. it annihilates the occupied MO pii- The operator basis {RI consists then on spin orbital level of operntors of different types, such ;ts (the subscriptsj, A_(+ [) are used for occupied unoccupied orbit&)
orbit&
other than 9;: G. b for
(1) external single substitutions if Q,Qj
-
ai Q,;
single substitutions from Us QiQr - Q:QQ; (3) internal singIe substitutions i(3)
Qi Qi
-
Qifaiir
(4) e.xtermd double substitutions Q,fabt’lx-Qj (5)
-
Q;Q&Q,
;
double substitutions Q;Q;QkQi
-
involving qi
-
(i,’ (in which ef is empty)_ The action of 7 on +, is the same as of 1 on +o. It is therefore recommended to “orthogomdize” 1 to 7 and to replace I by (I’) CompIementary Q;Q,Q&
single substitution
= Q;Qi - Q;Q;QrQi_
Now I’ acts only on G’ and the only operators acting on both @j, and @’ are those of type 4_ These. however, give (to second order in a) oniy unlinked terms, that do not contribute to A,!? and that can hence be ignored_ If $-, describes a closed-shell ground state_ we use a restricted Hartree-pock function with doubly occupied orbitais. It is then recommended to rep&e the substitution operators just given on spin orbital Ievcl by spin-adapted substitution operators on orbital level_ Instead of 7 types of opemtors one now gets IS different types, 5 of which correspond to external double substitutions and do not contribute to X FuII expressions of the remaining 13 operators will be given elsewhere. Here we limit ourselves to a pictorial classification. We assume that the orbital &- with &spin is ionized (a bar always designates P-spin). There are 3 types that act only on +,r (a) single substitutions from Zi: (b)doubIe substitutions from pi and pi to FQ and GQ; (c) double substitutions from tir-and Zr-to cs_ Z6 and GoTgbSeven types act only on +‘r (d)intemaI single substitutions (from $ to Zi): (e) conditional singie substitutions from “i (only if Zr is empty); (f) “relaxation” (substitutions from q,. to qb or Gi to Gb if hi iS empty); (g) “spin-polarization”; (11). (i). (j) semi-internal coupIing schemes_ fhree
double substitutions,
types act simultaneously
different
on +,, and ~1)‘:
Q;Q;QbQ,;
(6) semi-internal double substitutions t-r+-+ Q0Qi QkQ,- - Q,- Q,@& ; (7) conditionai single substitutions QzQTQpj
15 September 1979
CHEMICAL PHYSICS LETTERS
Q;Q;Q~=
_
The operators of type $,5 and 7 act only on $, (in which pi is occupied), those of type 3 and 6 only on
(k), @I)_ (m) doubie substitutions coupling schemes_ These 13 types of matrix elements tion the respective ed by a computer
involving ai_ different
of basis opemtors lead to 169 types BRs_ To avoid errors in their evahraformulas for the BRs were generatprogram. 113
Volume 66. number I
5_ Numerical resuks
4.Solution of the linear system The solution of the Liner system (I 1) is possible in a brute-force way- It is, however, better to proceed iterativeIy, namely to start with
04) and to obt& (01
BRRfR
l&*1)
the@*f) =
_-cp
R -
from c 8RS@’ S(*R)
until se!fconsistency_ One advantage of this scheme is tkt the matrix BRs need not be evJIuated as such (which would require a fuII twoeIectron integral trzcnsformation), but onIy tile VeCtOr ~S(_R$~fS which requires just one pzss through the i&egg! t;lpe_ The program used is based on the integral package described in ref. [I91 and the SCF part of the CEPA proo,clm described in ref_ [ZOl_
7s3P 9sSpld 9dp2d experiment [lZl
CEPA. IIsQ4dif
15 September 1979
CHEMICAL PHYSICS LETl-ERS
[Xl
Grcen’sfunction.s, 1ls7p7d [??I-
We have applied the method to several molecules and atoms_ In the present note we report the results for two representative systems, nsmefy H,O and Ne_ Tribe I ilhtstrates the dependence of the results on the basis of gaussian lobes- One sees that a sufficient number of poknization functions is necessary for the resuk to converge tolwrds the experimental vaIues_ We ~ISO compare with results from the literature and from experiment. In table 2 the total ionization potentials are decomposed into contributions of the model level (Koopmans energies) and various corrections_ The corresponding resuIts for Ne are given in table 3. One sees that the main corrections to the Koopmans values of ionization potent& of the vaience electrons come from the operators that describe “relaxation” (type (e) and (0) i-e_ that allow for the “Lremaining” orbit& in the ions to relax, and from the external double substitutions that involve yi (types (b), (c), (k), (n), (i-u))_ These types of contributions have the same order of magnitude, but opposite sign, so their contributions cancel to a large extent 1211 and the total correction is relatively small. The situ3tion is somewhat
lb2
lb,
3Zl
0.41937
O-49631
--0.67859
O-46154 O-46763 0.4698 0_45854 0.4730
0.54314 051936 0.545 f 0.004 OJ3950 0.5457
0.69999 0.70147 0.688 f 0.008 0.69286 0.6975
Ia1
19.79640 3)
18.821 19.83077
a) Basis 95Sp (uncontmcted) + J steep p-function. Table 2 Contributions to the ionization potenti& for water, basis: 9sSpZd (9Sp f up for lal)
IhopmsnsIp
=k..tfon
(types(e), (0)
spin po&rization &) externzddouble substitutions (b. c, k. P, m) semi-intennl double substitutions (lx. i, j) result
114
in au
-0.08895
-0.08099
0.71059 -0.05888
-0.01020 0.07346 -0.0095s G-46736
-0.00844 0.07451 -0.01368 054936
-0.00896 0.07447 -0.01576 010147
050290
0.57797
3055554 -0.77621 -0.02666 0.04166 0.00206 19.79640
Volume 66, number 1
CHEMICAL
TabIe 3 IonWtion potentktls for neon, basis: 9s5p2dlf (in 7-P Koopmans
fP
r&.kation spin pofarizztion external double substitutions semi-internal double substitutions result experiment 22.51
0.84465 -0.10687 -0.01081 0.08147 -0.01432 O-79413 0.7938
PHYSICS LJZTTERS
15 September 1979
References au>
2s
1.92392 -0.10668 -0.00942 0.04858 -0.07396 1_78245 1.7813
different for ionization from the core (Is)_ Here all corrections other than the relaxation are very sm3Il. such that relaxation dominates by far_ Sin&w observations bttve been made by other authors [22,23], based on different methods_ The comparison of the various contributions from different sources is not always easy. since these contributions cx! be defined differently. We have used the contributions $ fRAR that according to (is), sum up to the total Koopmans defect. The agreement of our results with experiment is of about the same quality as that of competing schemes. The remaining error is of the order of IO--2070 of the Koopmans error. It sItouId be mentioned that the metbod in the present form fails for certain ionizations from orbit& tbsr are between valence zrndcore orbit& (like 2a1 in fI20). The one-particle excitation is then probably n poor model excitation operator. The fact that most of the ionic states that are invoked in our calculations are strictly not bound states does otherwise not seem to cause problems-
Acknowledgement
El] J-P_ Xfialrieu,J. Chem_ Phyr 47 (1967) 4555. [2] D-P. Chong, F-G. Herring and D_ Mc\\rims, J_ Chem. Phys. 61 (1974) 78.. [3] V. KvamiCka and I. IiubnE, J. Chem. Phys. 60 (1974) 4483.
D.J. Rowe, Rev. Mod. Phys. 40 (1966) 153. IS] J. Simonsand WD. Smith, J_ #em. Phys. 58 (1973) 4899. 161 &f-F- Herman, D-L- Yeaser, K.F. Freed and V. McKay, Chem. Phys. Letters 46 (1977) i_ [7] J-D. Dolf and W.P. Reinhardt, J. Chem. Phys. 57 (1972) 1169.. JS] F. Ecker and G. Wohlncichcr, Theoret. Chim. Acta 25 _ (1972) 289. [V] L-S- Cederbaum, Tfxoret. Chim. Acta 31(1973) 239. [IO] L-S. Cederbaum and ‘w, Domeke. Advm. Chem. PhFs_ 36 (1977) 205. fill B-T_ Pickup and 0. Goscinski, Xfol. Phys. 26 (1973) 1013. [12] G.D.Pu~~andY.bhm,J.Chem. Phss. 60 (1974) 406% 1131 1. PaIdus. J_ &eli, 3L Saute zxd A_ L&orgue. Phys_ Rev_ Al7 (1978) SOS. 114j A.. Izfukhopzdhyay, TX. Moitm and D. 5f ukherjee, J_ Pbys_ B 12 (1979) 1. [l-5] SI.F_ Herman, D-L. Yeager and K.F. Freed, Chem. Ph>s_ 29 (1978) 77. (161 G-D. Purvisand 1’. dbrn, J. Chem. Phys. 65 (1976) 917. 1171 F. fhusdorfi: Ber- Verb- Konigt Sachs. Ges. Wi,s_ -sI~tb-Pbys_ Xl_ Leipzjg 58 (1906) 19. [18] %‘. Kutzeln&. --. in: i\Iodern theoretic& cbemistrv_VoL 3. ed. H-F_ Schaefer III (Plenum Pra% New York; i977) ’ p- 129. il9] R. Ahfrichs. Theoret. Chim. Actn 33 (1974) 157. PO1 R. Ahlrichs. H_ Lischb, V_ Staemmfer and IV- KutzeJp~, J. Chem. Phys 62 (1975) E25. Ial R-S. MulEken. J- Chim. Phys. 46 (1949) 497,675 f22] W- Meyer, Intern. J. Quantum Chem. S5 (1971) 341. P.S. Bztgus,Phys- Rev. 139 (1965) A619. g; W. van Niessen. G-H-F_Dierc~ksenand L.S. Cederbarn, J. Chem. Phys_ 67 (1977) 4124. 1251 C-E. Moore, Atomic Energy Levels, NatI. Bur. Std. CucuIm 467, Vol. 1 (1949). [4]
Tbis work was sponsored by Dents&e Forschungsgemeinscbsfr_ We thank Dr_ V. Staemmler and Dr. H. KolImsr for useful comments and suggestions.
115