Controlling orbital collapse from inside and outside a transition element

J. Phys. B: At. Mol. Opt. Phys. 31 (1998) 3557–3564. Printed in the UK

PII: S0953-4075(98)92554-8

Controlling orbital collapse from inside and outside a transition element J P Connerade† and V K Dolmatov‡ † The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK ‡ S V Starodubtsev Physical-Technical Institute, G Mavlyanova Str. 2, 700084 Tashkent, Uzbekistan Received 18 March 1998 Abstract. We report a study of bimodal behaviour of atomic wavefunctions for a transition element (Cr) and how this behaviour can be controlled and even turned into single-mode character either by (i) varying the effective nuclear charge for nonintegral values, or (ii) placing the atom in a spherical cavity of adjustable radius. Our conclusions have relevance to the emergence of valence instabilities for transition metals in the solid state. Also it is shown that the very existence of the bimodal wavefunctions provides an explanation for the frequently experienced instability of self-consistent field algorithms when a double-valley potential occurs, and suggestions are made on how to improve the numerical procedure for such calculations.

1. Introduction The phenomenon of orbital collapse in atomic species, arising when the atomic potential is of a double-well form, has important consequences for many properties of atoms as well as molecules, clusters and solids, and therefore the phenomenon has attracted much attention for some time (Griffin et al 1969, 1971, Connerade 1978a, 1997, Karaziya 1981). The expectation is normally to find orbitals entirely on one side or on the other of the potential barrier once a double-well potential has developed. However, another possibility exists which is much rarer, namely that the eigenfunctions should possess two maxima of not too dissimilar amplitudes, one on each side of the barrier. We refer to this as a bimodal solution, one mode being considered as an eigenfunction of the inner well and the other, of the outer well. Bimodal solutions were first found to occur naturally in the nf series of Ba+ by Connerade and Mansfield (1975). They were shown to account for an anomalous variation of the quantum defect which had initially been described by Saunders et al (1934) as a perturbation of a novel type, but had never properly been accounted for. Another important aspect of bimodal solutions is the ease with which the self-consistent field procedure may ‘flip’ from one solution to another, leading to ambiguities in the interpretation of Koopman’s theorem. In critical cases, it may become impossible to decide which is the ‘correct’ solution. Thus Band and Fomichev (1980) raised the question whether there might not be a coexistence of collapsed and ‘anticollapsed’ states for rare-earth elements, as a result of their Dirac–Fock calculations. Similarly, Connerade and Mansfield (1982) found two different solutions satisfying the usual tests for Hartree–Fock wavefunctions in Ba+ . It was suggested by Connerade (1982, 1983) that, while the coexistence of solutions is unlikely for free atoms, both solutions might possess physical significance for cases of mixed valence in solids, thereby suggesting c 1998 IOP Publishing Ltd 0953-4075/98/163557+08$19.50

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an alternative theory to parametrized approaches based on the Anderson impurity model. Independently, Schl¨uter and Varma (1983) reached similar conclusions on the basis of a Thomas–Fermi approximation. Band et al (1988) developed a quasiatomic Dirac–Fock model for intermediate valence in solids by using Wigner–Seitz boundary conditions. The particular model of Schl¨uter and Varma (1983) was criticized by Bringer (1983), but the question of the applicability of such quasiatomic models to atoms in the condensed phase remains open. They provide a natural explanation for the occurrence of valence instabilities in specific regions of the periodic table, and they also do not rely so heavily on adjustable parameters as alternative methods. In the light of the important consequences of the collapsed or bimodal-collapsed character of eigenfunctions for certain species, the possibility of controlling orbital collapse acquires significance. It was first found by Connerade (1978b, c) that the bimodal character of the nf series of Cs or Ba+ could be controlled by exciting a valence electron, and that, in favourable cases, the variation might be such as to make bimodal solutions appear and to enable the relative amplitudes of the inner and outer maxima to be controlled. Some experimental progress along these lines was made (Maiste et al 1980a, b, Lucatorto et al 1981). Until recently, the nf series of Ba+ seemed to be the only natural occurrence of the bimodal behaviour of eigenfunctions. Relatively recently, however, Dolmatov (1993a, b) (see also Dohrmann et al 1996) demonstrated the bimodal character of an excited 3d∗ (anti)collapsed orbital (arising from a 3p → 3d∗ transition) in free neutral Cr and also satisfactory control of such a behaviour by slightly varying the nuclear charge Z in selfconsistent-field Hartree–Fock equations, to account for the unique properties of the 3p absorption spectrum of free Cr. Actually, the theoretical basis for this procedure can be traced back to the foundations of the g-Hartree method. Dietz has shown (see Connerade and Dietz (1987) for details) that there is a family of central potentials represented as gV direct +(1−g)V exchange (V direct and V exchange are the direct and exchange parts of the atomic potential, respectively) which makes the action defined on the Lagrangian of quantum field theory (Itzykson and Zuber 1980) stationary. It turns out that solutions to this problem can be approximated by introducing a slight, fractional additional effective nuclear charge in the Hartree–Fock equations. The physical interpretation of this approximation is that the gHartree potential interpolates between the N-electron V N and (N −1)-electron V N−1 atomic potentials in an optimized manner (Connerade et al 1984, 1985) and therefore, roughly the same effect can also be achieved by a small variation of the nuclear charge Z. In this paper, we demonstrate one more way of controlling the bimodal character of collapsed eigenfunctions of the atom, namely by placing the atom inside a spherical cavity of adjustable radius R0 , with finite or infinite potential wells given by a potential step V0 ; this is achieved by altering the external boundary conditions for the atomic wavefunctions in self-consistent-field Hartree–Fock equations. This time the atom chosen for control is not a rare-earth element, but belongs to the 3d transition sequence. 2. Theoretical method Our calculations are performed in a spin-dependent self-consistent field scheme, namely, the spin-polarized Hartree–Fock approximation (SPHF) (Slater 1974), applied to calculations of Cr. Both free Cr and Cr confined in a spherical cavity are considered. For the latter case, the SPHF approximation is generalized and used to study confined atoms for the first time. The ground state of Cr is characterized by two half-filled 3d5 ↑ and 4s1 ↑ subshells (↑ (↓) denotes the upward (downward) spin direction), whose electrons all have a co-

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directed spin orientation (chosen explicitly as the upward direction), according to Hund’s rule. In the frame of SPHF, all otherwise closed subshells nl 2(2l+1) are divided into two subshells with opposite spin orientations, nl↑2l+1 and nl↓2l+1 , because of differences in the exchange interaction experienced by nl↑ and nl↓ electrons, caused by the imbalance in overall number of nl↑ and nl↓ electrons in the atom. Thus, the SPHF ground-state configuration of Cr looks as follows . . . 3p3 ↑ 3p3 ↓ 3d5 ↑ 4s1 ↑. The inner-shell excited configuration of Cr we consider in this paper is due to a 3p↓ → 3d↓ transition: . . . 3p3 ↑ 3p2 ↓ 3d5 ↑ 3d∗ ↓ 4s↑, in which a 3p↓ electron is excited to an initially unoccupied 3d∗ ↓ orbital; the 3d↑ and 4s↑ electrons are considered as just spectators to the inner-shell transitions. As follows from the SPHF approximation adopted in this paper (see also Amusia et al 1983), the single-electron radial wavefunctions (Pnlµ (r)) as well as single-electron energies (Enlµ ) acquire their spin dependence from the explicit dependence on z-projection (µ) of the electron spin, and can be obtained by solving the SPHF radial equation (Slater et al 1969) ! eff Znlµ l(l + 1) −1 d2 + Pnlµ = Enlµ Pnlµ (r). − (1) 2 dr 2 r 2r 2 eff The operator Znlµ , which can be regarded as the operator of an effective nuclear charge seen by an electron in the orbital nlµ, is defined by (   eff X Znlµ Z 1 n0 l 0 µ0 − Pnlµ (r) = − + V (r) + (Nn0 l 0 µ0 − δnlµ,n0 l 0 µ0 ) Y0 n0 l 0 µ0 ; r r r r n0 l 0 µ0 )   1 nlµ Nnlµ − 1 X k c (l0; l0) Yk nlµ; Pnlµ (r) − 2l r r k>0   X n0 l 0 µ0 Pn0 l 0 µ0 (r) Nn0 l 0 µ0 ck (l0; l 0 0) . (2) −2δµ,µ0 Y nlµ; k [(4l + 2)(4l 0 + 2)]1/2 r r k,n0 l 0 6=nl

Here atomic units are used; Nnlµ is the number of electrons in the subshell nlµ; the summations extend over all occupied states n0 l 0 µ0 in the atom; the coefficients ck represent products of three spherical harmonics; the functions Yk are defined as  Z ∞ k  1 n0 l 0 µ0 r< Pnlµ (r 0 )Pn0 l 0 µ0 (r 0 ) dr 0 (3) Yk nlµ; = r r r>k+1 0 where r< and r> represent the smaller and the greater of r 0 and r, respectively. We have also inserted an additional radial potential V (r) in (2), to simulate the situation of the atom confined inside a spherical cavity. It follows from (1) and (2) that radial wavefunctions (and also the phenomenon of eff seen by orbital collapse) can be controlled by changing the effective nuclear charge Znlµ an electron. eff ‘from the inside’, is to run a set of calculations of orbitals by One way to change Znlµ inserting different values for the nuclear charge Z into SPHF equations (1) and (2). The other, opposite, way we are going to consider in detail later in this paper, is a perturbation ‘from outside’, i.e. applying a spherically symmetric pressure to the outermost boundary of

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the atomic species by placing it in the centre of a spherical cavity of radius R0 with infinite potential step V0 : ( 0 if r < R0 (4) V (r) = V0 if r > R0 and letting Pnlµ satisfy the boundary condition Pnlµ |r=R0 = 0. In this work, the SPHF equations were solved within spheres of different radii R0 and a potential step V0 of −10 au was applied at r = R0 . The consistency of our method was tested (i) by checking that our solutions reproduce the exact hydrogenic results of Sommerfeld and Welker (1934), (ii) by checking that, for R0 → ∞, our solutions tend to those of the free atom and (iii) by checking that, for a given value of R0 , the total energy is reasonably stable for further increases in the height of the potential step. 3. Results and discussion 3.1. Controlled collapse as a function of Z In figure 1, we show the binding energies of the 3d↑, 4s↑ and 3d∗ ↓ electrons, together with the total energy of the system, plotted as a function of Z, the atomic number, considered as a nonintegral quantity (see the previous section). Notice the variations in binding energy (especially for the excited 3d∗ ↓ orbital) which are indicative of orbital collapse, and the sensitive dependence on even very slight changes in Z. Notice how the 3d5 ↑ and 4s↑ binding energies vary as a function of Z, and how this variation is counteracted by the variation for 3d∗ ↓. This behaviour will be further commented on below. Next, we show, in figure 2, the behaviour of the 3d∗ ↓ excited orbital. This is a separate orbital for the excited electron, because the 3d5 ↑ group is a spectator group already present in the ground-state configuration before the 3p↓ electron is excited.

Figure 1. Binding energies for the 3d5 ↑, 4s1 ↑ and 3d∗ ↓ electrons as well as the total energy E total of free Cr as a function of the nuclear charge Z.

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Figure 2. A set of radial wavefunctions P3d∗ ↓ (R) of the excited 3d∗ ↓ orbital (arising from the 3p↓ → 3d↓ transition) of free Cr for different values of the nuclear charge Z, as marked in the figure. Zones X and Y are circled for the sake of clarity.

The most noteworthy feature of figure 2 is the fact that there exist two classes of solutions to the SPHF equations, characterized by two different zones (marked X and Y ) in the figure, through which, to a good approximation, all the curves of a given family must pass. These two zones allow us to distinguish between the two classes of solution. In a double-well potential, the complete solution is made up from eigenfunctions of each individual well joined smoothly at some boundary between the two wells. The curves may thus be joined either to the right or left of the maximum in the eigenfunction of the outerwell. Since, for high Z, the outer electron tends to be attracted inwards, the wavefunction obtained by matching the curves to the right of the maximum (type 1 solution) are favoured, whereas the curves obtained the other way (type 2 solution) are favoured at lower Z. An interesting question, however, is what happens in between these two situations, i.e. under circumstances in which the two eigenfunctions are joined close to the maximum of the outermost one. It is very difficult to obtain SCF solutions in this parameter range, as the algorithm used to optimize the wavefunction becomes unstable around this zone. The reason for this is that the trial function can ‘flip’ from type 1 to type 2 between iterations. Indeed, we conclude from our study that current self-consistent field algorithms are inadequate in such a situation, and that new procedures are required for critical double-well potentials. Rather than allow the solutions to flip between type 1 and type 2, one should construct a complete map for each type of solution independently. A future possibility might be, in order to inhibit the instability, to construct a new algorithm by making small changes in the trial functions close to either of the X or Y , and larger changes elsewhere, so that it remains either a type 1 or a type 2 solution throughout the calculation. A convenient factor for this purpose would be (r − a)2 (r + a)2 which tends to 1 when r → 0 and when r → ∞ but is zero when r = a, and a is taken as the radius rX or rY . If part of the correction applied to the trial function is multiplied by ξ , the effect should be to inhibit ‘flipping’ in the unstable range. However, such a procedure would be unsuitable in this paper, because it would effectively force the wavefunction to be of type 1 or type 2, whereas our purpose is to show without biasing them in any ξ=

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way, that these bimodal solutions arise naturally. Consequently, we were unable to obtain convergence in the range between Z = 24.045 and 24.050 where the solution suddenly flips from type 2 to type 1. As the calculations show, altering the nuclear charge turns out to control orbital collapse in much the same way as exciting a valence electron to a Rydberg state (Connerade 1978b, c). 3.2. Controlled collapse inside a cavity Another way of controlling orbital collapse is to place the excited atom within a spherical cavity (Connerade 1997), which is easily achieved by altering the external boundary conditions and solving the Hartree–Fock equations within a sphere of given radius. In a sense, altering the nuclear charge and placing the atom in a spherical cavity can be regarded as two opposite forms of perturbation, one being the most internal and the other, the most external to the atom. Atoms in spherical cavities were considered very early in the development of quantum mechanics. A hydrogen atom placed inside a spherical cavity with impenetrable walls was considered by Sommerfeld and Welker (1938), who showed that, from the s states of the free hydrogen atom, exact solutions could be obtained. Even at this early stage, they recognized the connection this problem has to the theory of condensed matter. More recently, various applications in solid state physics were reviewed by Jask´olski (1996). Connerade (1997) used confined atoms in the context of ion storage to treat controlled orbital collapse in many-electron atoms contained in a spherical cavity, and showed that, because of orbital collapse, the ground configurations of these atoms depend on the size of the cavity. In figure 3, we show how the 3d∗ ↓ wavefunction of the 3p↓ → 3d∗ ↓-excited configuration of Cr behaves when placed in a spherical cavity consisting of a potential step of height 10 au. We have found by calculation that the solutions are stable in energy for a given radius as the height of the step is increased, once levels of about 10 au are reached. The graph shows 3d∗ ↓ wavefunctions for different radii. This figure should be compared with figure 2 above. Note how the same bimodal character of the wavefunctions

Figure 3. A set of wavefunctions P3d∗ ↓ (R) for the excited 3d∗ state (arising from the 3p↓ → 3d↓ transition) of confined Cr for different values of the radius R0 of a spherical cavity, as marked in the figure. Zones X and Y are circled for the sake of clarity. Note that, as was pointed out in the text, the outer zone Y spreads out, simply because of the phase shift of the outer eigenfunction produced by the presence of the outer cavity wall.

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Figure 4. Binding energies for the 3d5 ↑, 4s1 ↑ and 3d∗ ↓ electrons as well as the total energy E total of confined Cr as a function of the radius R0 of a spherical cavity.

emerges, and that orbital collapse can equally well be controlled by using an external cavity or by varying the nuclear charge. Indeed, much of the character of the functions in figure 2 is preserved in figure 3. Even the inner zone X occurs at approximately the same radius in both figures. The outer fixed zone Y is not as well preserved in figure 3, simply because of the phase shift of the outer eigenfunction produced by the presence of the outer cavity wall. Again, we were unable to obtain convergence in a critical range between R0 = 11 and 14. In figure 4, we show the binding energies of the 3d and 4s electrons and the total energy of the configuration. The significant point to note is the ‘atomic swing’ effect (Connerade 1997) in which the binding energy of the spectator spin-up valence electrons is decreasing, while the binding energy of the excited 3d∗ ↓ orbital at first increases and then decreases as the cavity radius is reduced. As a consequence of these opposing motions, the total energy of the configuration remains remarkably steady in the same range. Notice also the faster change of binding energy in the range 11 < R0 < 14, which happens to be precisely the range in which computations become very difficult and we failed to obtain convergence, essentially for the reasons given above.

4. Conclusion The calculations we have described demonstrate that orbital collapse may be controlled either from the centre, by altering the binding strength of the inner well, or from the outer reaches of the atom, by confining the outer well. This conclusion gives some insight into the observed difference between the spectra of free atoms and of the corresponding solids. Indeed, in a solid, the transfer of an electron from the inner well to the outer reaches of an individual atom can arise from a combination of two effects: first, as outer electrons are delocalized into the conduction band, the effective nuclear charge appears to be altered. Second, since the atom is contained inside a Wigner– Seitz cell, there is also an external effect due to the confining cavity. It is, therefore, not

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surprising, just by analogy with the effects of an external cavity or of a fractional nuclear charge as considered in this paper, that, for example, the spectrum of metallic chromium should bear very little relation to the spectrum of the free atom, although in cases where complete atomic orbital collapse occurs, giant resonances persist with very similar properties from the free atom to the solid (Sonntag et al 1969). Another application of the methods we have described arises when an atom is trapped inside a spherical fullerene cage. The simplest model for this situation is a sphere of charge, whose potential can be represented in a similar way to the model we have described. Indeed, we are currently extending our method to provide a simple description of endohedrally captured atoms. Acknowledgments This research was supported by the Royal Society of London under a Joint Research Agreement. JPC is grateful to colleagues from the Uzbekistan Academy of Sciences for hospitality at the Starodubtsev Institute (Tashkent) where this research was carried out. References Amusia M Ya, Dolmatov V K and Ivanov V K 1983 Sov. Phys.–JETP 58 67–72 Band I M and Fomichev V I 1980 Phys. Lett. 75A 178 Band I M, Kikoin K A Trzhavskovskaya M B and Khomskii D I 1988 Sov. Phys.–JETP 67 1561 Bringer A 1983 Solid State Commun. 46 591 Connerade J-P 1978a Contemp. Phys. 19 415 ——1978b J. Phys. B: At. Mol. Phys. 11 L409 ——1978c J. Phys. B: At. Mol. Phys. 11 L381 ——1982 J. Phys. C: Solid State Phys. 15 L367 ——1983 J. Less-Common Met. 93 171 ——1997 J. Alloys Compounds 225 79 Connerade J-P and Dietz K 1987 Comment. At. Mol. Phys. XIX 283 Connerade J-P, Dietz K, Mansfield M W D and Weymans G 1984 J. Phys. B: At. Mol. Phys. 17 1211 Connerade J-P, Dietz K and Weymans G 1985 J. Phys. B: At. Mol. Phys. 18, L309 Connerade J-P and Mansfield M W D 1975 Proc. R. Soc. A 346 565 ——1982 Phys. Rev. Lett. 48 131 Dohrmann Th, von dem Borne A, Verweyen A, Sonntag B, Wedowski M, Godehusen K, Zimmermann P and Dolmatov V K 1996 J. Phys. B: At. Mol. Opt. Phys. 29 4641–58 Dolmatov V K 1993a J. Phys. B: At. Mol. Opt. Phys. 26 L585–8 ——1993b J. Phys. B: At. Mol. Opt. Phys. 26 L393–8 Griffin D C, Andrew K L and Cowan R D 1969 Phys. Rev. 177 62 Griffin D C, Cowan R D and Andrew K L 1971 Phys. Rev. A 3 1233 Itzykson C and Zuber J B 1980 Quantum Field Theory (New York: McGraw-Hill) Jask´olsky W 1996 Phys. Rep. 271 1-66 Karaziya R I 1981 Sov. Phys.–Usp. 24 775 Lucatorto T J, McIlrath T, Sugar J and Younger S M 1981 Phys. Rev. Lett. 47 1124 Maiste A A, Ruus R E, Ruchas S A, Karaziya R I and Elango M A 1980a Sov. Phys.–JETP 51 474 ——1980b Sov. Phys.–JETP 52 844 Saunders F A, Schneider E G and Buckingham E 1934 Proc. Nat. Acad. Sci. USA 20 291 Schl¨uter M and Varma C M 1983 Helv. Phys. Acta 56 147 Slater J C 1974 The Self-consistent Field for Molecules and Solids (New York: McGraw-Hill) Slater J C, Mann J B, Wilson T M and Wood J H 1969 Phys. Rev. 184 672–94 Sommerfeld A and Welker H 1938 Ann. Phys., Lpz. 32 56 Sonntag B, Haensel R and Kunz C 1969 Solid State Commun. 7 597