Estimation of electronic correlation energies and binding energies for molecules composed of first-row atoms

ChemicalPhysics North-Holland

151 (1991)

159-167

Estimation of electronic correlation energies and binding energies for molecules composed of first-row atoms K. RoSciszewski *, M. Chaumet and P. Fulde Max-Planck-Institutjir Festtirperforschung, W-7000 Stuttgart 80, Germany Received 25 June 1990

The interatomic and intra-atomic correlation energies are calculated for a number of molecules composed of first-row atoms. This is done by starting from an ab initio SCF calculation and by treating the correlations with the help of local operators. The main result is the derivation of simple analytic formulae which tit rather well the results of the numerical computations. They depend only on the type of the chemical bond, the bond lengths and the Mullikan populations of the different atoms.

1. In~odIlction

Over the past few years considerable progress has been made in computing the correlation contributions to the ground-state energy of molecules. For examples see refs. [ 1- 15 1. Thereby the main emphasis has been to obtain as accurate results as possible and to study how they depend on the size of the basis set, on the special method of treating correlations etc. Comparatively little effort has been devoted to reproduce the computational findings in terms of simple rules and to understand their physical basis. Rules of this kind are desirable because they would enable one to estimate the effects of correlations on binding energies or other quantities without performing demanding calculations. An attempt in this direction was made in ref. [ 16 ] where the total correlation energy of different molecules was broken up into a number of approximately additive contributions. For each of these contributions analytical expressions were found which fitted a large number of data. However, these calculations started from semiempirical INDO (intermediate neglect of differential overlap) -like self-consistent field (SCF) calculations in which correlation calculations were implemented by means of the local ansatz (LA) method. Because of ’ On leave of absence from: Institute of Physics, Jagellonian University,

PI-30059

0301-0104/91/$03.50

Krakow, Poland.

the use of a minimal basis only interatomic correlations could be obtained in this way. Intra-atomic correlations require a larger basis set. They had to be included by an “atoms in molecules” type of approach. The calculations were put on a much sounder basis when for a number of hydrocarbon molecules the semiempirical SCF calculations were replaced by ab initio SCF calculations using larger basis sets [ 171. The results were encouraging. The various simple rules for interatomic correlation contributions were confirmed within an accuracy of IO- 15%. Regarding intra-atomic correlations a direct comparison was not possible. The estimates within the atoms in molecules approach refer to a (nearly) complete basis set, while the calculations based on the ab initio method were performed with a basis set including per atom one set of polarization functions only. This leaves a scaling factor between the two types of results. The aim of the present paper is to extend the calculations to molecules containing not only H and C but also B, N, 0 and F atoms. The goal is to derive simple analytic expressions again for the interatomic and intra-atomic correlation energy contributions. Only correlations between valence electrons are considered. The basis sets that are used are of triple-zeta plus one set of polarization functions quality. The paper is organized as follows. In section 2 a brief outline of the computational procedure is given including the necessary computational details. The

0 1991 - Elsevier Science Publishers B.V. (North-Holland)

K. RoSciszewski et al. / Energies for molecules composed offirst-row atoms

160

results are presented in section 3. The effect of correlations on binding energies is discussed in section 4. The remaining section 5 contains a brief summary of the results and the conclusions.

E,, are added up, and when, on the other hand, all the (60,) are used simultaneously. The {80ujlnterare local operators. Except for minor modifications they agree with the operators 0, = nit

2. Computational method

nil

=w,,

=Si’S,

The calculations were performed by starting from SCF calculations in a basisJ;( r) of Gaussian-type orbitals (GTO) and by treating correlations with the help of local operators. The details of the method (the local ansatz method) are found in refs. [ 14,15,19]. They are briefly summarized as follows. The correlation energy EC,, of the ground state of themolecule is written in the form

L,

= (60” IHI SO,) >

(2.1)

where H,,, = H- HsCF denotes the residual interactions that are not contained in the SCF part of the Hamiltonian HS,--. The round brackets are defined by the bilinear form (~I~)=(%I~+~I~,)‘,

,

(2.2)

where I G+,) is the ground state of HscF and the index “c” indicates that the cumulant of the expectation value must be taken. Expression (2.1) corresponds to a CEPA-0 approximation. The characteristic feature of the LA method is the choice of the operators {SO,} that describe the correlations. They have been discussed in detail in refs. [ 14,15,18]. We mention only briefly that the set of operators (80,) is decomposed into (80,) = {SOv}inter@{8Ou}intra,where the two sets of operators {8O,}i,e, and {GOv)intradescribe interatomic and intra-atomic correlations, respectively. Interatomic correlations are those that can be obtained from a properly chosen minimal basis set while intra-atomic correlations are the remaining ones, that is they require a larger than minimal basis set for their description. Although both sets of operators are coupled with each other when the correlation energy is calculated, the coupling is very small. Therefore, practically the same results are obtained when on the one hand the sets of operators {8Ov}inte, and IgOv)intraare used separately and the results for

1

v=

(&A

(2.3)

,

with % =

1 d

ni,,

nio

=

bf,b,,

;

(2.4) The three components of o are the Pauli matrices. The operators bf, (bi,) create (annihilate) electrons in local orbitals gi(r). The latter can be expressed in terms of the basis functionsh( r). Before we summarize their construction, which has been discussed in refs. [ 14,15 1, we want to mention that the {60,} are obtained from the {0,) by subtracting the zero- and one-particle excitations that are generated (in addition to the two-particle excitations) when the 0, act on I 0). This can be done either by formal subtraction or simply by forbidding contractions within the (0,) operators when expectation values are calculated with them [ 141. As far as the functions gi(r) are concerned, their number is one for each H atom, three for a B atom and usually four for the atoms C, N, 0 and F. These functions are orthogonalized to the 1s core states and Liiwdin orthogonalized with respect to each other. They agree essentially with the atomic sp hybrids aligned in the bond direction. As far as an H atom is concemeed, gi(r) nearly agrees with an atomic s function. Next the set of operators (6Ov)intmis discussed. It consists of all local excitations of the form {O,}c*{~~~,(A)~~~~(A)~~~,(A)~i,(A)} *

(2.5)

The operators u,,(A) are obtained by expressing the field operators vg( r) in terms of basis functions, that is

The index A runs over all the atoms that are also the

K. Rodciszewskiet al. /Energies for moleculescomposedoffirst-row atoms

centers of different GTOs. After subtracting from the {0,) the zero- and one-particle excitations the are obtained. When applied to I tPo) they {sou}intm describe focal two-particle excitations (atomic correlation operators) [ 191. One additional condition must be imposed though, on these operators. All those local two-particle excitations, which are already taken into account when interatomic correlations are treated must be excluded from the set of operators in order to prevent double counting. In the {60v>intra appendix it is shown how this selection of two-particle excitations can be related to the conventional configurations used, for example, in a CI calculation. The above theory is applied to a number of molecules composed of H, B, C, N, 0 and F atoms. The SCF part of the calculations is performed with the help of the Columbus program package (see ref. [ 10 ] for the Karlsruhe version of this). The following basis sets are used: for each H atom a 4s (2 11) basis according to Huzinaga plus a set of p polarization functions with exponent 0.75 [ 2 1,221; for the other atoms a 9s (51111), 5p (2111) basis according to Huzinaga, plus one set of d-polarization functions with exponents 0.388 (B), 0.6 (C), 0.864 (N), 1.154 (0) 1.496 (F), respectively [21,22].

contributions from o and x bonds. The second term cl.+.(Ai) contains those correlation energy contributions from lone pairs (1.~.) that can be described by a minimal basis set. They are small as compared with the intra-atomic contributions. Therefore, it is not a serious problem that the interatomic part of the lonepair contributions is not treated well within the local ansatz. For an improved treatment one would have to redefine the functions gi(r) when lone pairs are present (see ref. [ 16 ] ). The calculated interatomic energies of the different molecules can be fitted rather well by the following two analytic expressions: c(A-H)=

-0.12nAnn( 1+ 1.66Ad) eV,

The first expression describes the interatomic correlation energy of a bond formed by an H atom with a first-row atomic A, while the second expression describes that of a bond formed by two first-row atoms. The nA, nn are Mulliken gross population numbers for valence electrons. The Ad are deviations of the actual length of a bond from an average bond length d,,, that is (A-H) ,

Ad(A-B)=d-d,(A-B)

A total of 33 molecules are analysed. They are listed in table 1 together with their molecular skeletons. The older results for 11 hydrocarbon molecules [ 17 ] supplement the data. The results for the different correlation energy contributions and their tits by simple analytic expressions can be summarized as follows.

,*FA,,

WA,)

+

$

&l.p.(Ai)

(3.4)

respectively. The values for Ad that have to be put into equation (3.3) are in angstroms. Listed in table 2 are the values of do that have been used here. The parameters mABequal 1, 2 and 3 for single, double and triple bonds, respectively. The following values are used for the lone-pair contributions er+ (Ai) q.,(O]sp2)=-O.l07eV,

~i.~.(F]sp)=-0.279eV. The total interatomic correlation energy &interof a molecule is assumed to be of the form =

,

ctP.(0 I sp3 ) = - 0.176 eV ,

3. I. Interatomic correlation energy

&inter

(3.2)

&(A-B)= (0.031-0.061nzAB)nAnB( 1+ 1.55Ad) eV. (3.3)

Ad(A-H) =d-d,, 3. Results

161

3

(3.1)

where Ai and Ai are pairs of atoms forming chemical bonds. The energy E(A,-A,) is the correlation energy of the bond that is formed by the atoms Ai and AP It can be a single, double or triple bond. In distinction to ref. [ 16 ] this energy is not further decomposed into

(3.5)

Lone-pair contributions to the interatomic correlation energy of N, C (e.g., in CH*), B (e.g., in BH) and of poorly localized or defined lone pairs (e.g., in CO) are neglected when fitting equation (3.1) to calculated.values Of Eintcr(this may lead to errors in some cases; see the discussion given below ) . A comparison of the computed data and the data obtained by applying the analytic expressions (3.2)(3.5) is given in table 3 for a number of molecules.

162 Table I Correlation energies E,, presented is a comparison Molecule

K. RoSciszewski et al. /Energies for molecules composed offirst-row atoms

of the valence electrons for molecules with other ab initio data Skeleton

Hz0 Hz02

o-o

HNO

N-O

CHzG2

o=c-0

CH20

c=o

CH30H

c-o

NHZOH

N-O

NzH.,

N-N

CzHaNH if:

c-c NHzCN

N-C&N

I”\

CHzNz

CHzNH

N=N C=N

CH,NH*

C-N

CzNz NZ

N=C-C=N N=N

HCN

C-N

NH3

NzHz

N=N

NzG

N-N-0

co

cm0 o-c=0

HCNO CHzNz CH,CO

C=QN=0 C-N=N c-c=0

E_

(ax.)

composed

of first-row

‘)

talc.

anal. form

-0.220 -0.227 -0.426 -0.392 -0.381 -0.362 -0.522 -0.524 -0.344 -0.338 -0.341 -0.361 -0.367 -0.395 -0.379 -0.370 -0.375 - 0.456 - 0.424 - 0.490 -0.449 -0.460

-0.227

-0.319 -0.325 -0.343 -0.356 -0.565 -0.320 -0.326 -0.322 -0.302 -0.310 - 0.279 -0.301 -0.199 -0.196 - 0.207 -0.354 -0.349 -0.346 -0.533 -0.559 -0.301 -0.308 -0.501 -0.501 -0.504 -0.471 - 0.454 -0.470

-0.322

-0.416 -0.368 -0.534 -0.348

-0.362 -0.391 -0.362 -0.433

-0.453 - 0.446

-0.338 -0.562 -0.336

-0.305

-0.201

-0.355

-0.518 -0.300 - 0.498 -0.495 -0.487 -0.455

atoms as obtained

from the present theory. Also

Basis set

Method

Ref.

TZ+P’ TZ+P TZ+P’ 6-31 G** TZ+P’ 6-31 G** TZ+P’ DZ+P TZ+P’ TZ+P DZ+P TZ+P’ DZ+P TZ+P’ 6-31 G** TZ+P’ 6-31 G** TZ+P’ DZ+P DZ+P TZ+P’ TZ+P’

LA CEPA LA MP4 LA MP4 LA MP4 LA CEPA MP4 LA MP4 LA MP4 LA MP4 LA CI MP2 LA LA

I b’

TZ+P’ 6-31 G** TZ+P’ 6-31 G** TZ+P’ TZ+P’ TZ+P 6-31 G* TZ+P’ TZ+P TZ+P 6-31 G* TZ+P’ TZ+P TZ+P TZ+P’ TZ+P 6-31 G** TZ+P’ 6-311 G** TZ+P’ DZ+P’ TZ+P’ DZ+P TZ+P’ TZ+P’ TZ+P’ DZ+P

LA MP4 LA MP4 LA LA CEPA MP4 LA CEPA CI MP4 LA CI CEPA LA CEPA MP4 LA MP4 LA MP4 LA MP4 LA LA LA MP4

I

[311 I 1321 I 1321 I 1331 I [411 [331 I [331 I ~321 I [321 I 1341 1341 I I

1321 I ~321 I I 1311 1351 I 1311 1311 [351 I [81 181 I [361 ~321 I 1371 I [331 I 1331 I I I 1331

K. RoSciszewskiet al. /Energies for moleculescomppsedoffist-row atoms

163

Table 1 continued Molecule

E,,

Skeleton

FzNH

F-N-F

F,BH

F-B-F

CzHF

F-C-C

FNO

F-N-G

BH3

F-O

FOH

. F-F

F2

HF BH

(a.u.)

‘)

talc.

anal, form

-0.604 -0.505

- 0.629

-0.495 -0.415 -0.415 - 0.434 -0.583 -0.543 -0.108 -0.074 -0.120 -0.438 -0.373 -0.464 -0.434 - 0.368 - 0.232 -0.185 -0.067 -0.088

-0.528 -0.478 -0.580 -0.102

-0.439 - 0.460

-0,241 -0.059

Basis set

Method

Ref.

TZ+P’ 6-31 G*

LA MP2

I

TZ+P’ 6-31 G* TZ+P’ 6-31 G** TZ+P’ 6-31 G TZ+P’ 6-31 G* TZ+P TZ+P’ 6-31 G* TZ+P’ TZ+P 6-31 G* TZ+P’ 6-3lG* TZ+P’ TZ+P

LA MP2 LA MP4 LA MP4 LA MP2 CPA LA MP4 LA CEPA MP4 LA MP4 LA CEPA

I

1381

1381 I [391 I 1401 I 1381

[81 I [351 I 1311 1351 I 1351 I 181

‘) Abbreviations: talc., results of the present computations; anal. form, value of E,,,,, as obtained by using the empirical formulas (3.1)(3.6); DZ, TZ, double and triple zeta; P, polarization function (Id for each first-row atom, lp for hydrogen). The basis set TZ+P’ used in ref. [ 17 ] does not ditTer from TZ+ P for all practical purposes. ‘) I refers to the present paper. The entries in the table correspond to equilibrium geometric configurations of the molecules.

Table 2 Values of “average” in eqs. (3.4)

equilibrium

bond lengths 4, (in

A) as

used

A-H bond

&(A-H)

A-B bond

$(A-B)

B-H C-H N-H G-H F-H

1.19 1.08 1.00 0.95 0.92

F-B F-C F-N F-G F-F

1.31 1.30 1.40 1.45 1.42

A-B bond

&(A-B)

A-B bond

&(A-B)

c-c c-c c-c C-G co N-N N-N N-N

1.53 1.34 1.20 1.39 1.21 1.45 1.24 1.10

C-N GN C-N N-O N-G G-G

1.47 1.25 1.15 1.45 1.21 1.47

On the whole, equations (3.2)-( 3.5) reproduce hntcr within an accuracy of a few per cent. Exceptions are cyclic three-atom rings (e.g., in C2H4NH) where the bond angles deviate strongly from the optical ones for sp* or sp3 hybridization (differences up to 9%)) and furthermore molecules that have delocalized bonds as well as large charge transfers and poorly defined lone pairs (e.g., N20, CO, CO*, H&N,). Here the differences are up to 10-l 5%. The fits are least accurate for certain boron and fluorine compounds like FzBH or F,NH, in which large charge transfers take place. Here the values obtained from equation (3.1)(3.5) may deviate by up to 20% from the calculated ones. Furthermore, for NzO (N = N= 0) the parameter values mm=25 and mNo=1.5 are used [2325 1. Similarly, for CO a value of m, = 2.5 is chosen [ 241 for the fit. These half-integer values are the result of strong bond delocalization and are in fact approximations. In order to determine the degree of bond delocalization more accurately one would have

164

K. RoSciszewski et al. /Energies for molecules composed offrst-row atoms

Table 3 Comparison of the interatomic correlation energies for various molecules in their equilibrium configurations as obtained from the LA and by using analytic formulas (3.1)-( 3.6). Energies are in atomic units Molecule

HrD Hz% HNO CH202 CHsO CH,OH NH,OH NrH4 C2H,NH

Skeleton

o-o N-O o-c-o c=o c-o N-O N-N N /\ c-c N-C-N C

CHrN2 CNzHz

CHrNH CH,NHr GN2

/\ N-N C=N C-N N-C-C=N

H2C20

c=c=o

NZ HCN NH3 NsHr

N=N C=N N=N N=N=O c=o O-GO C=N-O C-N=N

N20

co CO, HCNO CHrNz

Deviation (%)

einter(au) talc.

anal. form.

- 0.050 - 0.097 -0.119 -0.148 -0.116 -0.106 -0.101 -0.109 -0.164

-0.049 -0.097 -0.113 -0.148 -0.118 -0.106 -0.102 -0.108 -0.151

1.9 -0.5 -4.8 0.1 1.8 -0.2 0.9 -0.3 -8.5

-0.166 -0.172

-0.160 -0.164

-3.4 -4.3

-0.121 -0.118 -0.251 -0.157 -0.131 -0.134 -0.061 -0.125 -0.193 -0.099 -0.155 -0.181 -0.185

-0.125 -0.116 -0.238 -0.159 -0.140 -0.130 - 0.060 -0.131 -0.176 - 0.092 -0.139 -0.174 -0.204

2.8 -1.6 -5.2 1.5 7.0 -3.2 -0.6 4.6 -8.8 -6.8 - 10.4 -4.0 10.2

to perform a population analysis as suggested by Ahlrichs [ 24,25 1. To keep the fitting as simple as possible we have used the Mulliken analysis despite some known shortcomings.

for boron compounds for which a Mulliken population analysis is questionable.

3.2. Intra-atomic correlation energy

By adding up einterand eintmas deduced from eqs. (3.1)-( 3.6) the total correlation energy Em= is obtained. Listed in table 1 is E,, for 33 different molecules. The geometric data were taken from refs. [ 26,271, respectively. The correlation energies calculated by means of the analytic formulae (3.1)(3.5) are compared with the numerical results obtained from the local ansatz and from other methods, that is CEPA and/or MP4 calculations. These values usually agree within l-2% with the directly calculated ones. The largest deviations occur for strongly

The numerical results for the intra-atomic correlation energy of an H atom or an atom A of the first row can be fitted well to the following expressions: G”,~~(A)= -O.l07nieV, Eimra(H)= -0.254nh

eV .

(3.6)

This is seen from figs. 1 and 2, respectively. Again, the deviations are largestfor three-centre bonds and

3.3. Total correlation energy

165

K. RoSciszewskiet al. /Energies for moleculescomposedoffirst-rowatoms

I

I

when the gross population of one atom is overestimated (by the Mulliken analysis) it must come out too small for others in order to match the fixed total number of valence electrons. The different correlation energy contributions are correspondingly overestimated, and underestimated, and the errors partially cancel. All in all, one notices that the analytic expressions give good agreement with the elaborate ab initio results.

I

I

I

-6 2

-5-

3

-4 -

g -3 (j- -2 _

3.4. Comparison with semiempiricalformulas

-1 -

2

I

I

I

I

I

3

4

5

6

7

I

“A Fig. I. Intra-atomic correlation energy (valence electrons only) of B, C, N, 0 and F atoms labeled A as functions of nA,that is the gross population of valence electrons when atom A is part of different molecules and when different geometric configurations of the molecules are considered (there is a total of 377 data points). Solid line: e,,(A) = -0.107n: eV.

-0.4

I

I

I

I

Eintra (H) z-0.254

I

I

1

t$,

-0.35

2 -

-0.3 -0.25 -

I

0.6

0.7

0.8

0.9

1.0

In ref. [ 16 ] analytic expressions for various correlation energy contributions were derived, which were based on semiempirical SCF calculations, that is by applying-an INDO scheme. Interatomic correlation energies were calculated by the local ansatz while the intra-atomic correlations were estimated from an “atoms in molecules” type of approach. The latter cannot be compared with the present ones because these estimations referred to nearly complete basis sets. However, the results for interatomic correlations can be compared with those of the present calculation that are based on ab initio SCF calculations. Thereby it is found that they agree within roughly 10% for C-H, C-C, C=C and CW bonds. For N bonds agreement is satisfactory for double and triple bonds, but unsatisfactory for single-o C-N and N-N bonds. This may be the result of an overestimation of the kinetic energy or hopping integrals for large bond lengths by the INDO method [ 28 1.

4. Binding energies of molecules

1.1

“H Fig. 2. Computed values of the intra-atomic correlation energy (valence electrons only) of a hydrogen atom in different molecules as a function of nu, that is the electron population on the atom (there is a total of 457 data points). Solid line: .r+,,&H) = -0.254& eV.

deformed bonds (up to 5%). The only exception is BH, with a deviation of 16%, for reasons pointed out before (no accounting for the lone pair on B). It is noticed that in total the deviations are smaller for E_ than they are for the single contributions. This is the result of a partial cancellation of errors. For example,

The correlation energy contribution to the binding energy of a molecule is given by &0* = &inter + C [Gma(&)-ECr(Ai)l Ai

>

(4.1)

where E&(Ai) is the correlation energy for the valence shell of the free atom A. The energy E&(A) must be calculated with the same basis set and by the same method as employed for the calculation of E,,. An open-shell SCF+LA package is at present being developed, but is not yet available. Therefore a CEPA0 program was used instead [29]. For the basis set given in section 2 the following values of &Am(A)

166

K. RoSciszewski et al. /Energies for molecules composed offirst-row atoms

are found for B, C, N, 0 and F atoms: - 1.912, -2.351, -2.748, -3.799, -4.872 eV. Previous calculations have shown that the LA misses approximately 5% of the correlation energy that can be obtained from a given basis set. We did not correct for these missing per cents because there is an additional overall correction arising from the limited size of the basis set when a comparisoli with the experimental values is made. The discrepancy between the calculated correlation contribution to the binding energy and the “experimental” values [ 301 amounts to approximately 30%. The latter are obtained when the estimated Hartree-Fock contributions are subtracted from the measured binding energies. This difference could be taken into account by a correction factor of (Y= 2.

Appendix: principle of configuration selection

The double substitutions generated by the local operators can be related to the standard configurations, i.e. those used, for example, in a CI calculation, as follows. First we introduce a complete set of one-electron orbitals by the creation operators c>,, of occupied molecular orbitals (MO) and CT,of unoccupied or virtual MO. The operators bf, in eq. (2.3) as well as the operators uf in eq. (2.3) can be expanded in terms of that complete set as N/2

b?o= 1 ui,&+ p=l

1

Vijctno >

n=(N/2)+1

N/2

(A.1 1 5. Summary and conclusions The main results of this paper are summarized in table 1. It demonstrates that the correlation energy of molecules consisting of H and of first-row atoms can be computed from simple analytic expressions. The results obtained from eqs. (3.2)-( 3.6) fit very well those of ab initio calculations. They hold for basis sets of triple-zeta plus polarization function size. The present findings suggest that approximate analytical expressions can also be found when larger basis sets are used or when the exact correlation energy is considered. The present approach should be set in contrast to earlier attempts of partitioning the total correlation energy (see ref. [ 421). The correlation energies of a number of molecules were decomposed in that paper into different bond contributions, but there was no attempt made to calculate correlation energies from simple analytic expressions.

Acknowledgments

The authors would like to thank Professor H. St011 for providing a set of atomic data on first-row atoms and for invaluable advice. We would also like to thank Dr. K. Vogel and Dr. A.M. OleS for many discussions and suggestions.

The corresponding expansions of the annihilation operators bin and ajo in terms of the C, c,, follows from eq. (A. 1) immediately. When, for example, the operator nitnil is taken (see eq. (2.3) ) then it generates the following double substitutions:

The terms that have been left out gene+ratesingle or no substitutions. Analogous expressions are obtained when the operators 0, given by eq. (2.5) are expressed instead in terms of substitutions. One notices that the local, non-orthogonal operators {0,} generate certain superpositions of conventional configurations ck,cf, c,, cpf I Cl+,) . These superpositions are best suited for generating the local contribution hole of an electron, and that is the reason why the local ansatz or local correlation operators are used.

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