Chemical Physics 61 (19Sl) 117-123 North-Holland Publishing Company

PSEUDOPOTENTIAL CALCULA’lTQNS ON Pz, P;, P4 AND P&f

U. WEDIG, H. STOLL and H. PREUSS Institut fr7r Theoretische Chemie der lJniL’ersit% 7000 Shzttgart 80, FRG

Received 23 Msrch 1981

Valence-only calclllations have been perfomed for Pz. P;. P4 and P,H-_ A semi-ioc4 semi-empirical pseudopotential was used, and valence-shell correlation was taken into account within the spin-density-functional formalism. Results are given for spectroscopic constants of Pz(‘Z.$ and P;(X’II,, A%:. F ‘Z:), for vertical ionization energies of P2, for the binding energy of PA and for its proton affinity. In particular, we discuss various directions of proton attack and the stability of the P,H’ complex with the proton in the centre of the p4 tetrahedron.

1. Introduction

In a recent paper [l], hereafter referred to as I, we determined and tested semi-local pseudopotentials for first-row (Li to Ne), second-row (Na to Ar) and third-row atoms (K, Ca). Core-valence correlation was included by adjusting the pseudopotentials to experimental energies of ions with a single electron. Corre- lation within the valence-shell was taken into account by using a spin-density-functional approximation.

The application of bur method seems to be most promising for molecules with more than two second- or third-row atoms, where the computational eRort in ab initio CI calculations is very large, if not prohibitive_ We already used our method for the determination of structures, binding energies and vertical ionization poten- tials of Na,- and K.-clusters up to n =8 [2,3]. From the study of clusters with essentially metallic bonding we now proceed to the investi- gation of clusters with covalent bonds, in par- ticular to that of P, clusters up to n =4.

For Pz near-I-IF results are available [4], which may be used to check the quality of oti pseudopotential. Calculations with non-local ab- initio pseudopotentials by Osman et al. [S] and by Nagy-Felsobuki and Peel [6] are not very

satisfactory as their orbital energies deviate from the near-HF values by ==1.5 eV and =4 eV, respectively.

For PJ SCF-all-electron calculations have been published [7, 81, but the basis sets are clearly insufficient here: with the larger one IS] P4 is predicted to be unstable with respect to 4 P. Valence-only calculations for PA have been performed by Osman et al. [S] and by Fluck et al. [Q]_ While the former are not expected to be too accurate (see above), a critical assessment of the latter is difi?cult, since no results for Pt have been given.

The P,H+ complex has been studied in ref. [9] by means of second-order perturbation theory, but it is not clear, if for the small separa- tions discussed (~1 A) second-order perturba- tion theory is even qualitatively correct.

In section 2 we give a short description of our calculational method. Section 3.1 contains results for orbital energies, vertical ionization potentials and spectroscopic constants of Pz in the ‘Ci ground state. Spectroscopic constants are also presented for some states of Pg (X ‘II,, A’C:, F’CZ). Binding energy, bond length and valence orbital energies for the P+ tetrahedron are given in section 3.2, together with corre- sponding results for P4H’-complexes of various geometries.

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2. Method

We use the following valence-shell model hamiltonian [lo]:

H mod = -4X Ai + 5 VLL I~.A:) i

0)

Here i, j denote valence electrons; A, p are core indices, and QA, Q, are the core charges. Our ansatz for the model pseudopotentials Vk, (rJ is:

V Q A. --nr> LX= mod=----e 4- 1 3[ e+Jrzfi. r. r I=0

Here the Pr are angular projection operatcrs, and I,,, is the maximum angular quantum num- ber in the core (i,,,= 1 for P). The parameters A, Q, BI, 61 in (2) (table-l) are adjusted to (experimentai) Rydloerg states of PJL_ Details of the fitting procedure are given in I. The accuracy of the fit is =iO-‘au for the two lowest states of s-, p- and d-symmetry. Other states are reproduced with errors ~3 x lo-’ au.

Correlation between valence electrons of different spins is approximated by a local spin- density functional [ll, 121:

Here E, is the valence correlation energy, p,“, pt are (partial) valence charge densities for a and p spin; .sc is the correlation energy per par- ticle for the homogeneous spin-polarized elec- tron liquid [13].

We have shown in I, that for the P atom our method reproduces the experimental value of

Table 1 Pseudqorentizl parametes for P

A = -2.044 0s = 2.125 B,=41.280 PO = 3.374 B1 = 22.734 & = 2.863

Table 2 Basis set for P,. P4

Symmetry type Eqment

s 1.391 0.352 0.125

P 0.337 0.098

bond mid-point S 0.35 P 0.35

the ionization potential with an accuracy of 0.1 eV. For the PH molecule we obtain a bind- ing energy which differs from that of an alI-elec- tron CI-calculation by only 0.3 eV.

Since the atomic pseudo-orbit&do not exhibit radia1 nodes, relatively small valence basis sets are capable of a remarkable degree of accuracy in pseudopotential calculations. In I we determined for the P atom a 3S/Zp GTO basis set (table 2) by means of a non-linear ieast- squares fit to the pseudo-orbitals of a 12s/9p basis. With the smalIer basis set, the total ener,T increases by only 3 x 10-* au, the ionix- ation potential changes by less than 0.1 eV.

It is clear from the work of Mulliken and Liu [4] that polarization functions are very impor- tant in Pz. The major part (2.5 eV) of the SCF binding energy for Pz is due to d- and f-contri- butions. In our calculations, we simulated polarization functions by s-p-sets in the bond mid-points (BFs). Such BFs have been tested for P2 in all-electron calculations by Carlsen [14]; they have been shown to be appropriate substitutes for d-functions. The exponent of the BFs (table 2) was determined by energy optimization for Pz at the experimental bond length.

3. Re&s

3.1. Pz, P2’

In tabble 3 we compare our orbit? energies and bond-length for Pz with pseudopotential results of other authcrs and all-electron near- HF values. The average-deviation of our orbital

119 U. Wedig er al. / Pseudopotential ralculations OH P2. P,‘, P+ and P,H’

Table 3 SCF orbital energies ei and bond length r, for Pz (VE: valence-only. AE: all-electron calculation)

Ref. Basis set -q(eV) r&Q

4=s 4% sflkz 2s”

This work GTO (3.0). 1BF VE 25.5 16.6 11.2 10.4 1.83”’ c41 STF (S/6/3/2) AE 24.8 16.4 11.1 10.3 1.85

ST0 3G+ Id VE 22.4 15.2 10.1 9.0 1.92 ST0 3G VE 20.6 13.4 6.9 5.9 1.84

=’ Including valence correlation: r, = 1.83 A.

energies from the near-HF values is 0.3 eV, the corresponding numbers for the previous pseudoptitential calculations are 1.5 eV and 4.0 eV, respectively_ In fig. 1 calculated poten- tial curves are shown for the ‘IX; ground state

2x P(‘S)

3.0 4.0 %.“.I

Fig. 1. Potential curves for various states of P2 and PC (- - -: experimental. -: calculated).

of Pz and the X %I,, A “Xi and F ?Z: states Pz, in comparison with experimental curves 1151; spectroscopic constants For these states are com- piled in table 4. The calculated binding energies are generally too small by more than 1 eV. This is partiaijy due to the fact that we have only a single bond function (BF) in our basis set; the contribution of this BF to the binding energy of Pz is 1.92 eV, while Mulliien and Liu [4] obtained an energy lowering of 2.52 eV from d- and f-polarization functions. Another source of errors is our density-functioiial approximation for the correlation energy [eq. (311. The correla- tion contribution to the binding energy of Pz is 2.0 eV in our approximation. From our experience with eq. (3) in previous work we conclude that this value is by 0.5 eV too small.

Our calculations indicate that the following assignments should be made for the ground and

Table 4 Binding energies 0,. bond-lengths r, and vibrational frequencies OJ, for Pz and P;. The FAculated values include valence correlation [SCFteq. (3)]; the oc are evaluated at the expzimental bond-lengths

D&V r, cri, w,,(cm’)

exp. Cl81 5.03 1.89 781 talc. x4 1.83 835

exp. [IS] 4.97 1.99 670 CrtlC. 3.6 1.89 674 exp. [lS] 4.7 1.89 733 CalC. 2.9 1.84 791 exp. [lS] - 1.8S 820*60 C&C. 1.57 918

Table 5 Vertical ionization potentials for Pz (in eV!. The calculated values include valence correlation (SCF+eq. (3))

Ionic state EW. Caic.

x ‘I-I, 10.62 10.3 A ‘X; 10.81 10.9 F ‘Z; 15.52 16.6

excited states of PE:

X ‘II, KKLL (4~~)‘(4~~,)‘(50~)‘(27i-,)~,

A ‘z; KKLL (4~s,)‘(4~“)“(5rr,)(27;-,)‘,

F ‘C: KKLL (4~~)‘(4~T,)(5~~)“(2~~)~-

Our assignment is in agreement with that of Bulgin et al. [15] with the exception of the notation, where we follow Huber and Herzberg [18j. Tn contrast to that, the configuration KKLL (4~~,)‘(40;)‘(5~~)(2~“~(2~~) was assig- ned to the F’EZ state in refs. lJ6, 171. From preliminary calcufations for that configuration, we find, however, that the bond length is 22.0 A (the experimental value for F ?-Cz is 1.85 A,*.

The (experimental) photoelectron spectrum in ref. [l5] shows three vertical ionization poten- tials for PZ with final (ionic) states X ‘II,, A 2x; and F ‘Zz. Calculated values are given in table 5. WhiIe the first two IPs are accurate to ~0.3 eV, the deviation from the experiment is markedly larger for F ‘ZZZ. As pointed out by Schirmer et al. [19], the (40U)-‘-ionization interacts strong19 with the (2rrJ-‘(5~~)-‘(2;r,)z) two-hole-one-particle excitation. This means that a one-particle approximation is inappropri- ate for ionization to F ‘YZz.

* The configuration con&s several Slate: determinants. Ziegler et al. [26] have shown that in density functional theory the energy of a contiguration can be evaluated from a wei&ted mean of single-determinant energies. ‘We have studied the potential curves of the most important determinants. AU of them had their minimum at r> ‘2.0 A. It is clear, however, that an explicit consideration of CI effects is necessary here, before final conclusions can be drawn.

3.2. P+ P.&T'

In table 6 we compare our calculated binding energy (with respect to 2 Pa), the bond length and valence orbital energies for the Pq tetra- hedron to previous theoretical work and to experimental data. Polarization functions are

Fig. 2. Valence-electron density in a u-,-plane of the P, tetrahedron. (a) P4: Basis set (3/Z). lb) P,: Basis set (3/2)-k BF. (c) PJIi Basis set (3/2)+BF, no functions at the pro- ton site. (Proton in edge position (X in fig. 3).)

U. Wedig et al. / Pseudopotential caicrrlations on P2, Pg. PJ and PJHC 121

Table 6 Binding energy AE, With respect to 2 Pz), valence orbital energies ei and bond-length r, for the PJ tetrahedron. AE and VE refer to a&eIectron and valence-only calculations respectively. Values in parentheses are calculated with “super-basis”. i.e.

using the P+ basis set for the P2 fragments

Ref. Basis set AE, (eV) -E; (eV) r,i&

4a, 5tz 5a, 6tz 2e

This work

c91

CSI

c71 ISI

GTO (3/2). Br GTO(2/1) GT0(3/2) GTO(3/Z/Z) ST0 3G ST0 3Gcld ST0 3G GTO( 1 l/7) [4, 31 Experiment Ref.

VE VE VE VE VE VE AE AE

1.29 (0.89)=’ 4.16 (2.65) 4.25 (0.48) 4.77 (2.52) 1.08

1.95

2.38 I201

z IncIuding valence correlation [SCF-ieq. (311.

” PES, mean vertical component of Jahn-Teller components.

important already for P2; for P4 fig. 2 shows the remarkable influence which they have on the valence electron density in the bonding region. Thus calculations for P4 without polarization functions should be regarded with some reserva- tion. In our caIcuIations, we simulate polariz- ation (d-, f-) functions by bond-functions (BFs), i.e. s-p sets in the mid-points of the edges of the P? tetrahedron. This is not optimal, of course; in fact, if we replace, for our calculated equilibrium geometries, the BFs by d-functions with the same exponent, the binding energy increases by 0.42 eV; a further improvement is to be expected if more than one d-set per P .

atom is used. Note that our exponent was optimized for Pz! Thus basis-set effects could account for more than G-5 eV of our binding energy error. It is satisfying, however, that basis super-position errors are much smailer than in the work of FIuck et al. f9], who obtained changes of up to 2 eV in the Pz total valence energy when using the PS (instead of the P2) molecular basis set.

Results of proton affinities and P.. Hr equilib- rium distances of the Pa tetrahedron are given in table 7. The geometry of the Pa tetrahedron was held fixed in these calculations. Using for P,HC the same basis set as in the P, c&uIations

Table 7 Binding energies (in eV) and (smallest) equilibrium PH’ distances rPH- (in A) of Hi approaching the P., tetrahedron from various directions. ?he bond length of the P, tetrahedron is fixed at the value given in table 6. All vaIues are calculated with va!ence correlation [SCF+eq(3)]

29.8 20.0 12.8 10.8 33.8 22.3 11.9 10.5 31.1 20.4 12.8 11.0 28.7 19.0 12.6 10.3 26.0 18.6 10.8 9.8 26.1 17.9 11.4 9.8 30.3b’ 20.0 11.6 10.4 31.5 21.0 12.5 Il.2

(14-19)” 11.9 10.4 [a

9.9 2.13z’ 9.4

10.2 8.9 2.25 9.0 2.33 9.1 9.7 2.36

10.5 r = r,,, 9.7 2.21

[211

Basis set TOP Edge Face

J% ‘PH’ &I Qw- EB re,i-

This work P: GTO(3/2), BF 2.8 1.6 3.2 1.4 2.3 1.6 P: GTO(3/2), BF: H+: GTO(3/1)=’ 6.9 1.4 6.9 1.5 3.6 1.7

Cal P: GTO(3/2) 3.5 - 1.1 - - - p: GT0(3/2/1) 4.3 - 2.6 - 3.5 -

” Basis for H: 3s (cf. ref. [24$ and lp (cf. ref. [25]).

(without basis-functions at the proton site), we obtain proton affinities, which are in .Fough qualitative agreement with the perturbational results of Fluck et al. [9]. The calculated values are 2-3 eV for various directions of proton attack. Thus the nucleophilicity of P4 seems to be remarkably low.

The fo!lowing considerations make it clear, however, that the basis set should be augmented by s- and p-functions at the proton site: The first ionization potential of P4 is smaller by 3.9 eV than that of the H-atom. Thus Pz+H is lower in energy than P,+H” by more than the proton binding energies calculated above. A significant amount of charge transfer is to be expected, consequently, in the ground states of the Pa.. _ El’ complexes.

The results with the augme&ed basis set show, that this is true indeed (table 7, fig. 3). The binding energies of the Pa.. .H’ comp!exes increase to 3.5-7 eV; they are comparable now to the proton aflinity of Nz (calculated: 5.5 eV [22], experimental: 5.1 eV [23]). For the Pa.. .H’ complex with H’ in top position, the equilibrium distance of the proton from the next-neighbouring P atom is very similar to the PH-distances in PH (1.43 L%) and PH3 (1.42&. The energies for top and edge positions of the proton differ by less than 0.1 eV, while the face position is higher in energy by I.3 eV. We expect edge and, in particular, face position to

FACE

Fig. 3. Binding-energy curves of H’ approaching the P4 tetrahedron from various directions. (x indicates the equilibrium position calculated without functions at the proton site.)

become slightly more favourable, if relaxation of the P-P bonds in the P4 tetrahedron is allowed for. Altogether, however, the regioselectivity of P4 against H* (and probably against an electrophilic agens in general) does not seem to be very strong. The lack of a mark&d nucleophiiic center at the P4 tetrahedron might be the reason for the difficulties to fix it experimentally as l&and in complexes.

A question remains: Is it possible for a pro- ton to penetrate the Pa face and to form a P4H+ complex with I-r’ in the centre of the tetrahe- dron? To answer this question, relaxation of the P-P bond lengths has to be taken into account_ Our calculations for P4HL with a central proton yield an optimized P-P bond length of 2.36 A (compared to 2.13 A for Pa). The binding energy (with respect to P4 + H”) is 1.7 eV, which is considerably smaller than that of the P4H’ complexes with the proton outside the tetrahe- dron. Additional calculations showed that there is only a very small energy barrier, if any, for the proton to leave the central position. Thus the proton in the center of the P4 tetrahedron does not seem to be stable, not &en meta- stable, as suggested by Fluck et al. [9].

Acknowledgement

The authors are gratefial to Dr. E. Golka and to F-X. Fraschio for supplying the programs used in the present calculations.

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