EOM/Propagator ionization potentials: Effect of an extended operator manifold on the predicted...

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XXVII, 145-157 (1985)

EoM/Propagator Ionization Potentials: Effect of an Extended Operator Manifold on the Predicted Valence-Shell

Ionization Spectra of Ethane, Ethylene, and Acetylene

JON BAKER* Department of Quantum Chemistry, Uppsala University, Box 518, S-75120 Uppsala, Sweden

Abstract

The orthogonalized direct diagonalization (ODD) method is used to predict the valence-shell ionization spectra for the series of hydrocarbons ethane (C,H,), ethylene (C,H,), and acetylene (C,H,) within an EoM/propagator formalism. Both (a) third-order and (b) higher-order (with h, terms in the excitation operator manifold) calculations are presented in order to illustrate the effect of the extended operator manifold on the predicted valence-shell spectra. It is shown that the major effect of the h, manifold is concentrated in its “occupied” part. Along with a general shifting of the main lines for outer valence ionizations to lower values (together with a slight reduction in pole strength), it is found that the effect of the h, manifold in the inner valence region is dependent on the system being studied.

1. Introduction

In an earlier paper [ l ] the direct diagonalization method [2,3] was used to obtain estimates for vertical ionization potentials (IPS) from the EoM/propagator matrix, with the excitation operator manifold utilized in this approach extended to include h5 operators (of the form a ~ a ~ a , a , a , ) . It was later shown [4] that, by a suitable orthogonalization procedure, this method could also be used to obtain “satellite structure,” enabling the main features of complete valence-shell ionization spectra to be predicted. The importance of the h5 manifold for the accurate prediction of features in the ionization spectrum has been demonstrated for hydrogen fluoride [5 ] . The purpose of this paper is to investigate the effects of the hS manifold on the predicted valence-shell spectra further for the series of hydrocarbons: ethane (C,H,), ethylene ( C2H4), and acetylene (C,H,).

The article begins with a brief discussion of the formalism (Sec. 2). No great detail is given as this has been adequately dealt with elsewhere [l , 51. In Sec. it is demonstrated that the major effects of the h5 operators are concentrated in the “occupied” part of the manifold, the “virtual” operators making very little contribution to the calculated ionizations. Making use of this, Sec. 4 presents two sets of calculations for the hydrocarbon spectra; the first being equivalent to a full third-order propagator and the second including the h5 manifold but without the virtual part. Section 5 comprises a discussion and conclusions.

* Present address: Research School of Chemistry, Australian National University, Canberra ACT 2601, Australia.

@ 1985 John Wiley & Sons, Inc. CCC 0020-7608/85/020145-13%04.00

146 BAKER

2. Brief Review of the Formalism

By adopting essentially the EOM approach [ 6 , 7 ] estimates for ionizations, w, are found by solution of the matrix equation

HC = wSC, ( 1 )

where

The excitation operator manifold h is limited to h, = {ap} , h3 = {a~a ,a , } and h, = { a ~ a ~ a , a , a , } operators, specified according to the recipe of Dalgaard [8] and spin adapted.

By a suitable redefinition of the manifold h and use of HF densities in the higher matrix blocks, it can be arranged that the S matrix in ( I ) is a unit matrix; thus (1) can be written explicitly as

Solution of (3) without the H5 blocks and with H F densities specified throughout is equivalent to the full two-particle-one-hole Tamm-Dancoff approximation (2p-h TDA) [9, lo]. By correcting the densities in the H” block to third order in RSFT, and in the H3‘ block to second order, results equivalent to a full third-order propagator can be obtained [3, 11, 121. Third-order calculations, together with calculations including the H5 blocks, are presented here. Expressions for all the matrix elements in (3) can be found in Appendix 2 of ref. [I].

The relevant information-eigenvalues and their corresponding eigenvectors-are extracted from the H matrix of (3) by orthogonalized direct diagonalization (ODD) [4]. It is considered that only those eigenvectors with “large” (main peaks) and “moderate” (satellite structure) cI components contribute to the predicted theoretical ionization spectra, in line with the simple “orbital picture” of ionization as being essentially the removal of an electron from a particular molecular orbital. The breakdown of this simple picture in the inner valence region is well established [13]; however, we still look for vectors with “moderate” c , components in this case also.

By using a modified version of the Davidson algorithm [14], starting with a ucii vector with unit cl component in the position corresponding to the ionization under examination, convergence to a vector representing the “primary ionization event” (i.e., a vector with a large cI component) is usually achieved. By restarting the procedure with a unit vector orthogonalized to the previously obtained eigenvector, convergence onto other vectors with “moderate” cI components can be attained. In this way a complete picture of the main features of the ionization spectrum can be gradually built up.

EO M/ PROPAGATOR I 0 N I ZATION 147

3. The hS Manifold

The h5 manifold can be divided (as can the hi and h3 manifolds) into an “occupied” part (consisting of operators of the form a ~ a ~ a i u , a k ) and a “virtual” part (consisting of operators of the form a ~ a ~ a , a b a , ) , where indices a, b, c rep- resent virtual and i , j , k occupied orbitals with reference to a single determinant HF ground state. After spin adaptation (to ensure that the ionic states are of doublet symmetry) there are five operators in each part of the manifold. The H5’ and H53 matrix blocks in (3) are set using HF densities (in which event H5’ = 0) and the H5’ block comprises a diagonal array of orbital energies only; within this approximation scheme it is found that, numerically, the major effect of the h5 manifold is concentrated in its “occupied” part. To illustrate this, Table I shows calculated principal ionizations (IP’S) for ethylene and water with and without the “virtual” part of the h5 manifold included. For ethylene the same basis and geometry were used as in Sec. 4, with a further truncation in the number of virtual orbitals used in the EOM to ten; for water an (10s6p2d 4- 5slp/5s3p2d + 3slp) Gaussian basis was used, with all five occupied and the lowest twelve virtuals in the EOM. Both basis sets were taken from Dunning [ 151. The calculations are full third order with the addition of the h5 manifold.

TABLE I. Principal ionizations (eV) to four decimal places for ethylene and water.”

ETHYLENE WATER

I P ( a ) f u l l h5 ( b ) occ. b5 I P ( a ) f u l l h5 (b ) occ. hy5

l a 2 9 8 . 8 1 5 1 298 .8151 l a l 5 4 8 . 1 9 2 1 5 4 8 . 1 9 2 1

l b l u 2 9 8 . 7 8 0 8 2 9 8 . 7 8 0 8 2 a l 3 2 . 6 2 5 4 3 2 . 6 2 4 2

Za 2 4 . 4 0 7 5 2 4 . 4 0 6 9 b2 18.8740 1 8 . 8 6 8 4

1 9 . 0 5 5 3 1 9 . 0 5 2 6 3 a l 1 4 . 7 8 2 2 1 4 . 7 7 7 8

1 2 . 4 7 0 2

9

q

2 b l u

b3u 1 5 . 5 9 3 1 1 5 . 5 9 0 7 b l 1 2 . 4 7 3 4

3a 1 4 . 6 0 6 2 1 4 . 6 0 3 6

1 2 . 6 5 8 3 1 2 . 6 5 4 5

1 0 . 1 8 8 8 1 0 . 1 7 7 5

q

29 b

b 2 u

a (a) Full h, manifold; (b) occupied h, operators only.

As can be seen from Table I the effect of the virtual operators is negligible for the outer valence region and diminishes to almost zero for core IPS, which agree exactly to four decimal places. It would thus seem very reasonable to leave out completely the virtual part of the h5 manifold, especially when it is noted that the H53 virtual matrix block is the biggest single block in the entire H matrix, requiring by far the largest computational effort for its construction.

148 BAKER

Calculations using the 2p-h TDA often involve truncation of the virtual part of the h3 manifold, since this usually has much less influence on the calculated IP’S than the occupied part [ 161. Another common approximation, when consider- ing valence-shell ionization spectra, is to leave out the core orbitals. The numerical effect of neglecting the virtual h5 operators is less than both of the above.

The unimportance of the virtual h5 manifold in the case of ethylene is clear from Table I ; corresponding calculations for the other two hydrocarbons considered here, ethane and acetylene, reveal a similar picture. Consequently, the calculations presented in Sec. 4, which include the h5 manifold, are with the occupied part only.

4. Results

As mentioned in Sec. 1, calculations were performed for the hydrocarbons ethane, ethylene, and acetylene. In an attempt to achieve roughly the same “quality” of calculation for each system, the same basis set was used throughout, an ( lOs6pld+5s lp/5~4pld +3slp) Gaussian basis taken from Dunning [15] [with ad ( C ) = 0.95 ; a,( H ) = 1 .OO], and the same number of virtual orbitals were retained in the EOM part in each case, namely, 24. All the occupied orbitals were utilized (including the core); thus, for ethane 9, ethylene 8, and acetylene 7 occupied orbitals, respectively. The integral, SCF, and integral transformation steps were done using the MOLECULE/EPSCF program package [17]. The geometry, molecular symmetry, total SCF energy and size of the excitation operator manifold h for each system are shown in Table 11.

TABLE 11. Geometry, molecular symmetry, total SCF energy, and size of excitation operator manifold for the hydrocarbons under study.

SYSTEM BOND LENGTH (A) BOND ANGLE SYMMETRY Escf S I Z E OF MANIFOLD

c-c C-H 6.H a c t u a l used ( h a r t r e e s ) n o b5 w i t h h_,

-79 .257295 7161 168585 C 2 H 6 1 .531 1 . 0 9 5 111.5‘

C 2 H 4 1 . 3 3 4 1 . 0 8 7 121.3’ D a h D 2 h - 7 8 . 0 6 1 1 5 5 6 1 7 6 118400

C 2 H 2 1 . 2 0 4 1 . 0 5 8 180‘ D d , D2h -76 .847264 5239 79411

D3d ‘2h

Before. discussing the results in detail, it is worth asking the question: what are the likely effects of the h5 manifold on the predicted ionization spectra? Previous work [I , 181 indicates that the main peaks will all be shifted to lower values, i.e., the estimated vertical ionization potentials will be less. Since the addition of h5 operators considerably enlarges the excitation operator manifold (see Table 11), the “pole strengths” (taken to be the square of the relevant c , eigenvector component) of the main lines are also expected to be less. The breakdown of the one-particle picture in the inner valence region in the 2p-h TDA and related schemes is considered to be due to near degeneracies between the

EOM/ PROPAGATOR IONIZATION 149

simple inner-valence configuration a k ( rCr,> and excited configurations such as ~ b a ~ a , l + ~ ) , where i and j are occupied outer valence orbitals and a is a low-lying virtual [13]. With an extended manifold there is the possibility of further near degeneracies between aklt,bo) and various h5 configurations; thus, the “inner- valence breakdown” is expected to be more marked-indeed an increase in satellite structure throughout the entire spectrum is likely.

A. Ethane

The energy profile for ethane is very sensitive to changes in the C-C bond length and the CTH bond angle near the equilibrium geometry [19]. In particular, the ordering of the SCF orbital energies for the two highest occupied MO’S depends very much on the geometry. There has been some controversy in the literature regarding the symmetry of the highest occupied MO, although it is now fairly well established that this is the leg orbital [19,20]. The geometry used in this work was taken from Kuchitsu [21]. The ionizations obtained for ethane, together with their corresponding pole strengths, are given in Table 111. Both third-order and extended manifold results are presented. Figure I depicts the theoretical “spike spectra.”

TABLE 111. Calculated valence shell ionizations (eV) and pole strengths (PS) for ethane.a

IONIZATION (a) t h i r d o r d e r (b) e x t e n d e d E x p t . I P PS I P PS r e f [203 r e f [22]‘

l e 1 2 . 4 5 0 . 9 3 2 1 2 . 2 7 0 . 9 1 6 1 2 . 4 1 2 . 2 5

1 2 . 9 8 0 . 9 2 3 1 2 . 7 0 0 . 8 9 7 1 3 . 5 1 3 . 3 5 3.3

1 5 . 2 4 0 .921 15 .01 0 . 8 9 8 1 5 . 4 1 5 . 4 5

2 1 . 0 2 0 , 8 5 6 2 0 . 1 8 0 . 7 8 4 2 0 . 4 2 1 . 0

2 a 1 q 2 4 . 7 9 0 . 6 4 6 2 2 . 5 4 0 . 0 1 1 24 2 4 . 5

9

19

l e u

2 a 2 u

2 5 . 3 1 0 . 1 2 6 2 3 . 8 7 0 . 5 2 9

2 3 . 9 9 0 .021

2 4 . 7 4 0 . 0 1 5

2 5 . 5 3 0 . 0 1 2 2 5 . 6 1 0 . 0 3 2

2 6 . 1 4 0 . 0 6 3

a (a) Third order; (b) extended manifold. *0.1 ev .

Both spectra show four main lines for the four highest ionizations, with no notable satellite structure. The pole strengths are all high for the three outer valence IP’S (around 0.9), and only for the lower 2u2,, IP is the pole strength reduced. This clearly indicates the validity of the “orbital picture” (see Sec. 2)

150 BAKER

20 15 30

ionization energy (eV)

Figure I . Predicted valence-shell ionization “spike spectra” for ethane. (a) Third order. (b) Extended manifold. Labeling: ‘le, * 3 a , , 31e,, 42a,,, ’ 2 s , ,

for these ionizations-a picture which persists in the extended manifold calculations, despite the huge increase in the number of operators involved.

For the inner-valence 2a,, IP however, there is a noticeable difference between the two spectra, with the extended manifold showing much more satellite structure, although there is still a discernible “main” peak. The third-order spectrum, on the other hand, shows only one 2a,, satellite, albeit a rather significant one. It should be noted at this point though, that ODD does not guarantee that all relevant eigenvectors of (3) will be found [4]; thus, for example, there may be one or two small 2a,, satellites for the third-order spectrum which have not been located. Similarly, and even more likely, for the extended manifold spectrum.

The ionic excited state contributing most to the third-order 2 a I , satellite is (1 e,)-’(aT,). For the extended manifold satellites several excited ionic configurations contribute in varying degrees; however, it is notable that all the significant contributary configurations are derived from the h3 part of the manifold, there being no single h5 configuration of weight more than about I YO. The h, operators

EOM/PROPAGATOR IONIZATION 151

would seem to produce their effect simply because there are so many of them; lots of small contributions rather than one or two major ones. This, of course, is not particularly surprising; on the contrary, it would indeed be surprising if any component of real significance were to come from the h5 manifold. In fact the total weight for any given satellite from the whole of the h5 manifold is probably less than 10%. This would appear to be a general feature, since the same holds for both ethylene and acetylene.

The lack of satellite structure in the ethane spectrum has been commented on previously in Ref. 16, where even the 2al, ionization showed outer-valence character, although this was probably due to shortcomings in the approximation scheme used (2p-h TDA).

In general, agreement with experiment for the main peaks is good throughout, with rhe exception of the 3a, , IP. However, the experimental spectrum itself is not easy to interpret directly; there is a broadband from about 11.5 to 14.5 eV, and the leg and 3 a , , peaks have to be separated by deconvolution [22]. Which of the two calculations give the best results depends on the experimental value taken for comparison; on the whole the third-order results are probably better, with an average error (comparing with the experimental value which agrees closest to the calculated IP) for the four IP’S (excluding the 3a,,) of 0.13 eV, as opposed to 0.19 eV for the extended manifold.

B. Ethylene

The results for ethylene are shown in Table IV and the two spike spectra in Figure 2.

The third-order ethylene spectrum shows much more satellite structure than the corresponding spectrum for ethane. Not only does the inner valence 2ag IP

show the familiar breakdown, but both the 2b1, and b3, IP’S also have satellite lines, although in both cases a definite main line exists. This satellite structure becomes more pronounced in the extended manifold spectrum, and indeed, both IP’S show signs of the breakdown normally associated with the inner-valence region, the b3, ionization markedly so. The inner valence IP itself falls to pieces in the extended manifold, there are a multitude of lines in this region of the spectrum, the largest with a pole strength of only 0.276.

In contrast, the three outer-valence IP’S are very “stable,” the extended manifold giving no additional satellite lines and only a slight loss in pole strength,

For the 2Bl, and B,, satellites the major ionic contributions come from the (3ag)-’( & , , - I ( b&) and the ( b2,)-I( b2,)-’( bf,) configurations, respectively. By comparing the vectors from the two sets of calculations, it can be seen that the 2bl, satellite at 20.43 eV in the third-order spectrum corresponds to the satellite at 17.77 eV in the extended manifold, while the satellite at 25.42 eV corresponds to the line at 22.61 eV. Similarly the b,, satellite at 17.84 eV shifts to 16.13 eV and the one at 22.91 eV occurs at 20.63 eV in the extended manifold spectrum. Clearly the effect of the h5 manifold on satellite structure is greater than its effect on the main line, both in terms of change of pole strength and shift of the

152 BAKER

TABLE IV. Calculated valence shell ionizations (eV) and pole strengths (PS) for ethylene.”

IONIZATION ( a ) t h i r d order ( b ) e x t e n d e d IP PS IP PS

1 0 . 2 9

1 2 . 8 2

3 a 1 4 . 6 5

1 5 . 9 8 1 7 . 8 4

2 2 . 9 1

b2u

2 9

9

b

b3u

1 9 . 3 0

2 0 . 4 3

2 5 . 4 2

2 b l u

2a 2 3 . 1 8 2 4 . 3 2

2 7 . 6 4

2 9 . 5 5

9

0 . 9 1 9

0 . 9 2 2

0 . 9 1 4

0 . 8 2 8

0 . 0 3 7

0 . 0 6 0

0 . 7 3 1

0 . 0 9 2

0 . 0 6 3

0 . 0 3 3

0 . 5 2 8

0 . 1 7 6

0 . 0 7 0

1 0 . 1 4

1 2 . 6 1

1 4 . 3 4

1 5 . 3 7

1 6 . 1 3

2 0 . 1 5

2 0 . 6 3

1 7 . 7 7

1 8 . 8 5 2 0 . 7 5

2 2 . 6 1

2 2 . 6 2

2 3 . 0 8 2 3 . 8 2

2 4 . 9 2

2 6 . 3 3

2 6 . 8 8

2 7 . 3 4

2 7 . 4 6 2 7 . 7 4

2 7 . 7 8

2 7 . 9 6 2 9 . 3 5

2 9 . 6 0

0 . 9 1 4

0 . 3 0 0

0 . 8 8 4

0 . 5 1 9

0 . 2 6 6

0 . 0 0 9

0 . 1 0 8

0 .061

0 . 6 1 8

0 . 0 2 2

0 . 1 3 8

0 . 0 9 8

0 . 2 7 6 0 . 0 1 0

0 . 0 4 4 0 . 0 4 8

0 . 0 6 2

0 . 0 6 7

0 . 0 3 5 0 . 0 1 5

0 . 0 1 2

0 . 0 8 4

0 . 0 1 1

0 . 0 1 2

1 0 . 5 I a

1 2 . 8 5 a

1 4 . 6 6 a

1 5 . 8 7 a

19.23’

23.65’

a (a) Third order; (b) extended manifold. Ref. [23]. ‘ Ref. [24].

ionization to lower values. No such direct comparison can be made with the 2a, satellites, since the additional breakdown is too marked.

Regarding agreement with experiment, the third-order results are excellent. Apart from the bzu IP, for which it is known that additional polarization functions are required in the basis for an accurate estimation [25], the accuracy of the calculated ionizations is very high. The average error for the four IP’S from b,, to 2blU is only 0.05 eV.

The extended manifold IP’S, however, are consistently too low. It is also debatable if there is any evidence from experimental spectra to support the degree

EOM/PROPAGATOR IONIZATION

15

153

516

4. I II1.1 ..

I

c .-

(a )

10

II I0

20

i 5

,5 l6 I; 30

6

20 30 Ionization energy (eV)

Figure 2. Predicted valence-shell ionization “spike spectra” for ethylene. (a) Third order. (b) Extended manifold. Labeling: ‘b,,, ’b,,, 33a,, 4b,,, ’2b, , , 62a,.

of breakdown predicted here, although this possibility should not be excluded, since very rarely are the observed bands sharp and they are normally very broad in the inner valence region.

C. Acetylene

Table V gives the calculated ionizations for acetylene and Figure 3 the corresponding spike spectra.

Beyond the expected shift of the lines to lower values and reduction in pole strength for the main peaks, there is very little additional structure in the extended spectrum. Even the 2ag inner-valence ionization retains the same character as in the third-order spectrum, namely, a splitting into essentially two lines, although the pole strengths are shared more evenly between the lines in the extended spectrum and the line separation somewhat reduced (from 5.2 to 4.48 eV). The situation here is very similar to that found for hydrogen fluoride (see Ref. 5).

154 BAKER

TABLE V. Calculated valence shell ionizations (eV) and pole strengths (PS) for acetylene.”

IONIZATION ( a ) t h i r d o r d e r (b) ex tended Expt .

IP PS IP PS r e f [26]

T U 1 1 . 0 2 0 . 9 2 2 1 0 . 8 4 0 . 9 0 4 1 1 . 4 3

1 7 . 0 3 0 . 9 1 0 1 6 . 7 2 0 . 8 7 8 1 6 . 7 5

2rr, 1 9 . 0 0 0 . 8 5 3 1 8 . 5 1 0 . 7 6 3 1 8 . 7 0

?¶

2 3 . 9 4 0 . 0 1 7 2 1 . 4 0 0 . 0 2 3

2 5 . 7 7 0 . 0 2 3 2 2 . 5 0 0 .058

2 3 . 9 6 0 . 5 7 5 2 2 . 5 7 0 . 3 5 8 2 3 . 6 5

2 8 . 7 6 0 . 3 0 7 2 7 . 0 5 0 . 4 1 2

3 0 . 2 7 0 . 0 5 7

r3

a (a) Third order; (b) extended manifold.

1 20 30

4

14

Ionization energy (ev)

Figure 3. Predicted valence-shell ionization “spike spectra” for acetylene. (a) Third order. (b) Extended manifold. Labeling: ‘rU, ’3aP ’b,, 4Zur


Typical experimental ionization spectra for acetylene [26] show a broad shake-up band around 27.5 eV. A recent high resolution ESCA spectrum, studied by means of monochromatized x rays [27], separated this broad satellite structure into two bands, centred at 26.8 and 28.0 eV. In both the third-order and extended manifold spectra only one line is present in this region, at 28.76 and 27.05 eV, respectively, although the extended spectrum does show an additional feature at 30.27 eV. The main excited ionic configuration contributing to this line is (2wU)-’( T~)-’( T:). There would thus appear to be an essential disagreement between the predicted and experimental spectra in this case, and also a contrast to the usual situation where the experimental spectrum is not sufficiently resolved to determine whether all the predicted satellites are present or not.

Overall, the extended manifold gives better agreement with experiment for the main lines than the third-order results, although both calculations predict a low value for the rU IP. Probably further polarization functions are needed in the basis to get a better estimate for this IP, as is the case with the b2“ IP for ethylene.

5. Discussion and Conclusions

The results presented here support the general comments made at the begin- ning of Sec. 4, namely, that inclusion of the h5 manifold results in a shifting of the main lines in the valence-shell ionization spectrum to lower values, together with a reduction (often slight) in the pole strength. This is the situation for the highest outer-valence ionizations. As one progresses towards the inner-valence region, however, the effect of the h5 manifold is dependent on the system being studied. For ethane, nothing additional happens until the inner-valence IP itself is reached, when there is a further breakdown and increase in satellite structure: for ethylene, some of the higher ionizations show the “inner valence breakdown” with the inner valence IP collapsing completely to a host of satellite lines; whereas for acetylene there is hardly any change in the overall pattern of the spectrum and no significant extra satellite structure.

In Ref. 16, the authors use the existence of low-lying excited states to explain the observed differences between the predicted 2p-h TDA spectra of saturated versus unsaturated hydrocarbons, the argument being that such states “encourage” the near degeneracies between single hole and other ionic configurations that are required for the breakdown of the orbital picture of ionization. Since unsaturated hydrocarbons have these low-lying states, their spectra show a considerable number of satellite lines : saturated hydrocarbons, which do not have low-lying excited states, consequently show much less satellite structure.

Although this argument can be used, at least in part, to explain the differences between the extended manifold spectra for ethane [Fig. l(b)] and ethylene [Fig. 2(b)], it cannot be used when comparing the ethylene and acetylene [Fig. 3(b)] spectra.

Overall, the effect of the h5 manifold on the predicted ionization spectra is perhaps best described as “variable.” Sometimes the results are an improvement over the standard third-order treatment (e.g., acetylene, hydrogen fluoride), other

156 BAKER

times apparently a disaster (ethylene). The greatest effect is likely to be in the inner-valence region, with a significant increase in the number of satellite lines a distinct possibility. It is difficult to test for this experimentally without access to “cleaner” spectra with much greater resolution than currently available.

One reason for the variable nature of the h5 results may be that the fourth-order terms so introduced are not being introduced consistently. Whereas the third-order treatment described here includes all third-order terms, the h5 manifold is just one of the sources of possible fourth-order terms (although it is likely to be the most important one). Fourth-order terms also arise from correcting the densities in the lower matrix blocks to higher-order, e.g., RSPT corrections to the HF density in the H33 block.

Some discussion of the likely effects of higher-order terms in the H matrix is given in Ref. 28 and a recent paper by Schirmer, Cederbaum, and Walter [29] details how the relevant terms can be included in a systematic manner so as to arrive at a full fourth- (or higher-) order scheme. As far as the h5 manifold is concerned, the only fourth-order terms not included here come from second-order terms in the H5’ block, and preliminary work (unpublished) has indicated that these terms are not important. Evidence has been given in this paper (Table I) that the virtual part of the h, manifold appears to have very little effect on the calculated IP’S, and so it would seem appropriate to concentrate on the lower matrix blocks if an attempt is made to attain a reasonably consistent fourth-order scheme. However, until more is known about the effects of other fourth-order terms, or until there is a more reliable procedure for including hS (or other operators) in a manner which will definitely improve the quality of the results, it seems best to use the simpler third-order theory for actual calculations.

Acknowledgments

Thanks to Professor 0. Goscinski and the Quantum Chemistry Group at Uppsala for their hospitality, Dr. H. Olofsson at the VAX for excellent computational facilities, J. A. Cogordan for assistance with some of the computations, and the Science Research Council (United Kingdom) for financial support.

Bibliography

[I] J. Baker, Chem. Phys. 79, 117 (1983). [2] J. Baker and B. T. Picaup, Chem. Phys. Lett. 76, 537 (1980). [3] J. Baker and B. T. Pickup, Mol. Phys. 49, 651 (1983). [4] J . Baker, Chem. Phys. Lett. 101, 136 (1983). [5] J . Baker, J. Chem. Phys. 80, 2693 (1984). [6] D. J. Rowe, Rev. Mod. Phys. 40, 153 (1968). [7] T. H. Dunning and V. McKoy, J. Chern. Phys. 47, 1735 (1967); 48, 5263 (1968). [8] E. Dalgaard, Int. J. Quant. Chem. 15, 169 (1979). [9] J. Schirmer and L. S. Cederbaum, J. Phys. B 11, 1889 (1978).

[lo] G. D. Purvis and Y. Ohm, J. Chem. Phys. 60, 4063 (1974). [ I 13 P. J~irgensen and J. Simons, J. Chem. Phys. 63, 5302 (1975). [12] M. F. Herman, D. L. Yeager, and K. F. Freed, Chem. Phys. 29, 77 (1978). [I31 J . Schirmer, W. Domcke, L. S. Cederbaum, and W. Von Niessen, J. Phys. B 11, 1901 (1978).


[I41 E. R. Davidson, J. Comput. Phys. 17, 87 (1975). [I51 T. H. Dunning, J. Chem. Phys. 53, 2823 (1970); 55, 3958 (1971). [16] L. S. Cederbaum, W. Domcke, J. Schirmer, W. Von Niessen, G. H. F. Diercksen, and W. P.

[I71 J. Alrnlof, et al., University of Uppsala, Sweden and Quantum Theory Project, University of

[I81 L. T. Redmon, G. Purvis, and Y. Ohm, J. Chem. Phys. 63, 5011 (1975). [I91 A. Richartz, R. J. Buenker, P. J. Bruna, and S. D. Peyerimhoff, Mol. Phys. 33, 1345 (1977). [20] J. W. Rabalais and A. Katrib, Mol. Phys. 27, 923 (1974). [21] K. Kuchitsu, J. Chem. Phys. 49, 4456 (1968). [22] S. Dey, A. J. Dixon, I. E. McCarthy, and E. Weigold, J. Electron Spectrosc. Relat. Phenom. 9,

[23] M. B. Robin, N. A. Kuebler, and C. R. Brundle, in Electron Spectroscopy, D. A. Shirley, Ed.

[24] M. S. Banna and D. A. Shirley, J. Electron Spectrosc. Relat. Phenom. 8, 255 (1976). [25] W. Von Niessen, G. H. F. Diercksen, L. S. Cederbaum, and W. Domcke, Chem. Phys. 18, 469

[26] R. G. Cavell and D. A. Allison, J. Chem. Phys. 69, 159 (1978). [27] J. Muller, R. Arneberg, H. Agren, R. Manne, P. A. Malmquist, S. Svensson, and U. Gelius, J.

[28] M. F. Herman, K. F. Freed, and D. L. Yeager, J. Chem. Phys. 72, 602 (1980). [29] J. Schirmer, L. S. Cederbaum, and 0. Walter, Phys. Rev. A 28, 1237 (1983).

Received December 23, 1983 Accepted for publication June 11, 1984

Kraemer, J. Chern. Phys. 69, 1591 (1978).

Florida, Gainesville.

397 (1976).

(North-Holland, Amsterdam, 1972).

(1976).

Chem. Phys. 77, 4895 (1982).

EOM/Propagator ionization potentials: Effect of an extended operator manifold on the predicted...

Documents

Transcript of EOM/Propagator ionization potentials: Effect of an extended operator manifold on the predicted...