Depurated Inversion Method for Orbital-Specific Exchange Potentials
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Depurated Inversion Method for Orbital–SpecificExchange Potentials
A.M.P. Mendez ∗ , D.M. Mitnik, J.E. Miraglia † August 30, 2018
Abstract
This work presents exchange potentials for specific orbitals calculated by invert-ing Hartree–Fock wavefunctions. This was achieved by using a Depurated InversionMethod. The basic idea of the method relies upon the substitution of Hartree–Fockorbitals and eigenvalues into the Kohn–Sham equation. Through inversion, the corre-sponding effective potentials were obtained. Further treatment of the inverted potentialshould be carried on. The depuration is a careful optimization which eliminates thepoles and also ensures the fullfilment of the appropriate boundary conditions. Theprocedure developed here is not restricted to the ground state or to a nodeless orbitaland is applicable to all kinds of atoms. As an example, exchange potentials for noblegases and term–dependent orbitals of the lower configuration of Nitrogen are calcu-lated. The method allows to reproduce the input energies and wavefunctions with aremarkable degree of accuracy. ∗ [email protected] † Instituto de Astronom´ıa y F´ısica del Espacio, CONICET–UBA, Buenos Aires, Argentina. INTRODUCTION
The successful idea of replacing a many–body, non–local interaction by an effective one–electron equation opened up the possibility of studying extremely complex systems withhigh accuracy. In this context, the success of the Kohn–Sham density–functional theory (DFT) began when crucial developments in its exchange–correlation terms gave the the-ory predictive power to compete with well–developed wavefunction methods . The impor-tance of the exchange–correlation potentials in chemical physics has been emphasized byBartlett . Exchange potentials are in general constructed by local approximations to thenonlocal Hartree–Fock exchange operator (i.e. the Slater potential , the optimized effectivepotential , the Krieger–Li–Iafrate and several others ).The atomic collision community, on the other hand, is also eager to accurately determineeffective one–electron local potentials which would allow to generate in a simpler way thewavefunctions of the particles interacting in a scattering process. In particular, we needto represent an orthonormal set of bound and continuum states to calculate the transitionprobabilities. This should include detailed nl –orbital potentials, a feature missing in mostof the standard density functional methods. Soft pseudopotentials like abinit or uspp cannot be used because they overlook the information of the internal region of the wavefunc-tions. The features of this region can play a very important role, such as the cusp conditionsin the processes of electron capture and ionization. In an attempt to meet the needs ofboth chemist and collisionist communities, we strove to obtain accurate and simple specific nl –orbital local potentials.How to determine central potentials from known electron wavefunctions and densities isa well studied subject in the DFT community . The extraction of the true Kohn–Shamexchange–correlation potential from near–exact electronic densities has been demonstrated,with particular reference to two–electron systems like He , He–isoelectronic ions , andH as well as exact soluble models (for example, an external harmonic potential as inFilippi et al ).Some other works start with a particular Kohn–Sham potential and solve the correspond-ing equations, obtaining the KS orbitals . Through inversion, they obtain a reconstructedKS potential, which agrees almost everywhere with the original one, except in some regionswhere huge oscillations arise. In some cases, the reconstructed potential may be distortedbeyond recognition . The same type of procedure was suggested many years ago byHilton et al. , in applications circumscribed to the calculation of photoionization processesof atoms , water and other molecules . These papers, in turn, refer to the earlierwork in atomic polarizability carried out by Sternheimer and Dalgarno and Parkinson .However, they focused on the final photoionization cross section results, and did not providedetails about the quality of the potentials and the wavefunctions they generated.Assuming the validity of the separation between exchange and correlation functionals, wewill focus here only on the calculation of the exchange contribution to the potential. SinceHartree–Fock does not include the correlations, our approach allows to obtain the “exact”one–electron local potential representing the exchange interactions. Strictly speaking, themethod does not rely on the KS inversion formula since the Hartree–Fock solutions were theones used for the inversion. That is, we solved a KS–type equation, but rather than havingKS–orbitals, we operated directly with the Hartree–Fock wavefunctions. For open–shell2toms, we were able to find orbital spin–polarized exchange potentials, this being crucial,for instance, to find the hyperfine coupling constants .However, this is not a simple task, and probably that is why the method presented herehas not been widely applied in the past. If the wavefunction has nodes, it will produce hugepoles in the potential. Moreover, even for nodeless states, the asymptotic decaying behaviorof the bound wavefunctions produces severe numerical difficulties, making the inversion op-eration intractable sometimes. In our method, a depuration procedure follows the inversion.This depuration implies, first, the annihilation of the poles. Then, a careful optimization ofthe potential which ensures the fulfillment of the appropriate boundary conditions.The work is organized as follows. Section 2 describes the method, which includes theinversion procedure (2.1), the potential depuration (2.2) and its further optimization (2.3).Section 3 presents the resulting effective potentials for the orbitals corresponding to theground states of different noble gases, including a thorough examination of the wavefunctionsgenerated by these potentials (3.1). The corresponding exchange potentials are discussed in(3.2), comparing the potentials for specific– nl orbitals with averaged potentials. Results ofthe same calculations for the Nitrogen atom are provided in (3.3). Atomic units are usedunless otherwise specified. THEORY
The direct inversion method
The radial part of the Schr¨odinger equation for an electron in a local and central potential is (cid:20) − d dr + l ( l + 1)2 r + V nl ( r ) (cid:21) u nl ( r ) = ε nl u nl ( r ) . (1)We assume the following hypothesis: If the wavefunctions u nl are replaced by the solutionsof an Hartree–Fock calculation u HF nl , then, the corresponding effective local potentials V HF nl that generate such wavefunctions should exist. Based on this we converted the HF methodinto a set of Kohn–Sham equations, whose solutions are the Hartree–Fock wavefunctions: (cid:20) − d dr + l ( l + 1)2 r + V HF nl ( r ) (cid:21) u HF nl ( r ) = ε HF nl u HF nl ( r ) . (2)The effective potentials given by, V HF nl ( r ) = V C ( r ) + V dir ( r ) + V x nl ( r ) , (3)are composed of the external potential V C (the Coulomb field of the nucleus), the direct(or Hartree) potential V dir (the electrostatic electron repulsion), and the orbital exchangepotentials V x nl . We have ignored the correlation term since the HF solutions do not include it.Since the solutions u HF nl are known (calculated numerically with the hf code by C. F.Fischer , and the nrhf code by W. Johnson ) we proceeded to directly invert the Kohn–Sham–type equations: V HF nl ( r ) = 12 1 u HF nl ( r ) d dr u HF nl ( r ) − l ( l + 1)2 r + ε HF nl , (4)3btaining the HF inverted potential V HF nl ( r ). Assuming a Coulombic–type shape, it is con-venient to define an HF inverted charge Z HF nl ( r ) ≡ − r V HF nl ( r ) . (5)The direct computation of (4) is known to pose serious numerical problems . First , thepresence of (genuine) nodes in the wave function to be inverted produces poles and unrealisticfeatures around them. This has led to the general consensus that the inversion method canonly be used for nodeless orbitals . Second , numerical rounding up of the exponential decayof the bound states hinders the corresponding inverted potential from having the physicallydesired asymptotic form. Moreover, there is a third problem at the very heart of the HartreeFock method: the exact solutions may have oscillations (and therefore, spurious nodes) inthe large–r or “tail” region of the functions. The existence of these spurious nodes in HartreeFock was already suggested by Fischer . This failure is not caused by the numerical schemebut it is inherent to the method. Probably, these nodes are surviving long–range exchangeeffects due to the non–local character of the Hartree–Fock wavefunctions: the behavior of aparticular orbital depends on all others. We have found the same spurious nodes at the sameplaces even using different numerical codes. As a general rule, the spurious nodes appearat very long distances, in regions where the amplitude of the wavefunction is very small.Therefore, their existence has no practical consequences, and they can be ignored in anygeneral Hartree–Fock calculation. However, this is not true as far as the inversion procedureis concerned, as we will discuss in the next section. Other examples where the presenceof orbital nodes (both formal and those in the tail region) can be problematic in inversionprocedures can be found in the literature (see for instance Peach et al. ). The depurated inversion method
The difficulties mentioned above make it very hard to obtain the correct V HF nl ( r ) potentialsusing the simple inversion formula given by Eq. (4). To overcome these troubles we havedeveloped a depurated inversion method (DIM) which optimizes the effective charges ratherthan the effective potentials. We managed to constrain any potential to have the rightboundary conditions by enforcing the effective depurated inverted charge to behave as follows: Z DIM nl ( r ) → (cid:26) Z N as r →
01 as r → ∞ (6)where Z N is the nuclear charge. Once the charge is determined at the boundaries, we canobtain a smooth analytic expression for Z DIM nl ( r ), fitting the Z HF nl ( r ) for the largest possiblerange, except in the neighborhood of the nodes. All this can be accomplished by imposingthe effective DIM charge to fit the following analytical expression: Z DIM nl ( r ) = X j α j e − β j r + 1 , (7)with Σ j α j = Z N − u HF2 s ( r ) of the ground state of the Kr atom (part (a)),4 -5 -2024 u s r (a.u.) Z s (a)(b) Figure 1: (a) Hartree–Fock orbital u HF2 s corresponding to the ground state of the Kr atom.It presents two nodes, a genuine one at r ≈ .
06 a.u., and a spurious one at 1 .
51 a.u. (shownin the inset). (b) Dashed line: The corresponding inverted effective charge Z HF2 s ( r ), spoiledby the presence of poles. Solid line: Depurated Z DIM2 s ( r ) effective charge.and its correspondent effective charge Z HF2 s ( r ) (dashed line curve, in part (b)). First, notethat the 2 s orbital has a genuine node at r ≈ .
06 a.u. which produces the first pole in theeffective charge, as shown in the lower graph. The node appears at a relatively low– r value,so the corresponding charge (see Eq. (5)) is not very sensitive to its presence. Therefore,it is very easy to eliminate the pole from the effective charge (by just erasing a few pointsaround this radius).All the bound wavefunctions decay exponentially beyond the last turning point r tp , de-fined as the position in which the energy equals the effective potential. At first glance, itseems that the turning point of u s ( r ) is located around r tp ≈ .
25, and from that point on,the wavefunction should start to decay exponentially. From the numerically point of view, r ≈ r tp is a good point to stop the inversion, since beyond there, the effective charge couldbegin to diverge. Thus, one might infer that by erasing the points belonging to the neighbor-hood of the first node, and by stopping the inversion about 10 r tp , the inversion procedurewill work well. However, the dashed curve in Fig. 1(b) shows a completely unphysical Z HF2 s ( r )resulting from the inversion. A very careful examination of the u HF2 s ( r ) orbital function ev-idences the presence of a spurious node at r = 1 .
51 a.u., in a region where the amplitudeof the wavefunction is less than 10 − times the maximum value (see the inset of Fig. 1(a)).Even though this node is completely innocuous for practical matters, it produces devastatingeffects in the inversion procedure, evidenced by the second huge peak in the Z HF2 s ( r ) curve(see Figure 1(b)). This pole is so big that it affects a broad vicinity and causes the abruptrising of the effective charge for r > . a priory there is no reason to suspect that a negligible oscillation in the tail of the wavefunction wouldproduce such a big drawback at small distances. Care must be taken then to discard thesekind of undesired effects. 5 .3 Optimization
The adjustment of the parameters α j and β j also requires carefull work. The key issue in thesuccessful approximation is the region chosen for the fitting: it has to be as large as possible,in such a way that Z DIM nl ( r ) overlaps the inverted Z HF nl ( r ) across a broad range, allowingan accurate fitting procedure, but discarding the points surrounding the nodes. Also, theinversion must be halted at a particular (as large as possible) r value, as soon as the amplitudeof the function is too small. Further on, the inversion procedure diverges. Another issue toconsider is the self consistency within the computer codes used in the calculations and theparticular code used to generate the input wavefunctions. To that end, we make sure thatthe same specific numerical grid is used, including the derivatives and integrals at the samepivots. The optimization procedure is completed by a number of iteration steps, in whichthe parameters are optimized to give accurate energies and wavefuntions.Most density functional approximation methods are based on a variational principle, min-imizing the density functionals according to energy (others are defined by density). Withoutunderestimating its importance, energy is only one of the many parameters that character-izes a quantum state. Different trial functions (having different forms) can produce, througha variational procedure, the same final energy. A simple example is given by Bartschat in which two different potentials (one having exchange, the other omitting it) led to produc-ing very similar and accurate energies of the Rydberg series in several quasi–one electronsystems. However, a further examination of these potentials shows large discrepancies inscattering calculations . Therefore, in addition to the energy criterion, we have included inour optimization method a variational procedure to reproduce accurately the wavefunctions.This is achieved by optimizing the mean values h /r i (which characterize the quality of thewavefunction near the origin), and h r i (probing it at longer distances). Furthermore, wedefined the quantity δ = 1 − R u HF nl ( r ) u DIM nl ( r ) dr R ρ HF nl ( r ) dr . (8)to determine the accuracy of the orbitals generated by the diagonalization of the DIM po-tentials and the original HF orbitals.The effective depurated inversion charge Z DIM2 s ( r ) corresponding to the 2 s orbital of theKr atom resulting from the optimization is shown –solid curve– in Figure 1(b). As seen in thefigure, both boundary conditions are fulfilled (at the origin, Z s →
36, and asymptotically Z s →
1, as stated in Eq. (6)). RESULTS
DIM Potentials, energies and mean values
The fitting parameters α j and β j defining the effective charges Z DIM nl ( r ) in Eq. (7) for thenoble gases Helium, Neon, Argon and Krypton, are given in Table 1. We have limited the α j and β j to six (about two per shell). For Kr, we would probably need two more sincethere are four shells involved. Having these effective charges, we built the correspondingDIM potentials V DIM nl ( r ). By solving the Schr¨odinger equation (Eq. (1)), we obtained the6able 1: Fitting parameters for the effective charge Z DIM nl ( r ) for He, Ne, Ar, and Kr, applyingEq. (7). nl α β nl α β He 1 s -0.31745 5.04372 Kr 1 s s s s p p s s p s d p s s p p u DIM nl ( r ) and the corresponding energies ε DIM nl . The comparison between the resultsobtained from the diagonalization of the Hamiltonian with the V DIM nl ( r ) effective potentialand the original Hartree–Fock orbitals are presented in Table 2. It is remarkable that withsuch simple analytical expressions for the potentials we were able to reproduce exactly thesame energies as the HF method. The only exception is the 4 p orbital of Kr, in whichboth calculations agree up to the fifth significant figure. The fitting procedure also allowsto reproduce the original HF wavefunctions with an outstanding degree of accuracy. Theagreement between the HF orbitals u HF nl ( r ) and the solutions u DIM nl ( r ) can be tested throughthe comparison of the mean values h r i and h /r i , and the computation of quantity δ definedby Eq. (8). The mean values agree in about 0 .
1% while the values of δ are about 10 − .Finally, we calculated the total energy for the ground state of each atom, by using thefollowing expression: E DIM = X nl (cid:20) ε DIM nl − Z ρ DIM nl ( r ) (cid:18) V DIM nl ( r ) + Z N r (cid:19) dr (cid:21) , (9)where the density ρ DIM nl ( r ) = | u DIM nl ( r ) | . The calculated energies E DIM are given in Table 2,together with the total energies obtained by the Hartree–Fock calculations. The comparisonshows a notable agreement between both calculations, at about 0 . The exchange potential
Orbital–specific exchange potentials can be obtained accurately by computing the non–localFock exchange operator. A first local approximation can be computed with the averageexchange charge density proposed by Slater . Another approximation, proposed by Sharpand Horton , consists in attaining a local potential that approximates the exchange operatorthrough a variational procedure that minimizes the energy. There are several other moreelaborated methods that allow us to obtain local exchange potentials . However, thesepotentials are rather difficult to put in a simple and smooth analytical expression, such asEq.(7).Due to the fact that the Hartree–Fock method does not take into account the correlations,our procedure allowed us to obtain in a rather direct way “exact” local orbital–dependentexchange potentials, V DIMx nl ( r ) = V DIM nl ( r ) + Z N r − Z ρ HF ( r ′ ) | r − r ′ | d r ′ , (10)where ρ HF ( r ) is the total density calculated with the u HF nl ( r ) wavefunctions. Figure 2 showsthe orbital–specific exchange potentials V DIMx nl ( r ) for the ground states of the four noblegases He, Ne, Ar, and Kr, calculated with the depurated inversion method DIM.In order to discuss our results, in Fig. 2 we plotted the optimized effective potential V xOEP ( r ) developed by Talman (black dotted lines) for the noble gases. It is well known thatthe oep method finds the potential which yields eigenfunctions that minimize the expectationvalue of the Hartree–Fock Hamiltonian. However, although very accurate, it always yields anenergy above the HF energy. For practical applications the oep potential works very well for8able 2: Total and orbital energies, mean and δ values for He, Ne, Ar and Kr atoms obtainedfrom DIM effective potentials (upper rows) compared with the original Hartree–Fock values(lower rows). E nl ǫ h r i h /r i δ He − . s − .
917 956 0 .
927 313 1 .
687 251 8 × − − . − .
917 956 0 .
927 273 1 .
687 282Ne − . s − .
772 447 0 .
157 491 9 .
621 450 2 × − − . − .
772 443 0 .
157 631 9 .
618 0542 s − .
930 391 0 .
891 336 1 .
640 769 5 × − − .
930 391 0 .
892 113 1 .
632 5532 p − .
850 410 0 .
967 755 1 .
430 252 6 × − − .
850 410 0 .
965 274 1 .
435 350Ar − . s − .
610 352 0 .
086 015 17 .
561 606 2 × − − . − .
610 350 0 .
086 104 17 .
553 2292 s − .
322 153 0 .
411 857 3 .
562 264 2 × − − .
322 153 0 .
412 280 3 .
555 3172 p − .
571 466 0 .
375 269 3 .
449 283 9 × − − .
571 466 0 .
375 330 3 .
449 9893 s − .
277 353 1 .
426 944 0 .
967 005 9 × − − .
277 353 1 .
422 172 0 .
961 9853 p − .
591 017 1 .
668 648 0 .
817 928 5 × − − .
591 017 1 .
662 959 0 .
814 074Kr − . s − .
165 467 0 .
042 441 35 .
483 699 5 × − − . − .
165 468 0 .
042 441 35 .
498 1522 s − .
903 081 0 .
187 181 7 .
924 967 2 × − − .
903 082 0 .
187 256 7 .
918 8302 p − .
009 784 0 .
161 695 7 .
874 355 3 × − − .
009 785 0 .
161 876 7 .
868 4293 s − .
849 466 0 .
537 875 2 .
644 610 2 × − − .
849 466 0 .
537 802 2 .
637 5563 p − .
331 501 0 .
542 133 2 .
530 080 2 × − − .
331 501 0 .
542 627 2 .
522 7753 d − .
825 234 0 .
550 922 2 .
276 713 4 × − − .
825 234 0 .
550 880 2 .
276 9404 s − .
152 935 1 .
630 081 0 .
808 453 1 × − − .
152 935 1 .
629 391 0 .
804 1884 p − .
524 186 1 .
950 193 0 .
675 555 3 × − − .
524 187 1 .
951 611 0 .
669 2199 .01 0.1 1 10 r (a.u.) -2-1.5-1-0.50 V x (r) OEP1s He r (a.u.) -10-8-6-4-20 V x (r) OEP1s2s2p Ne r (a.u.) -20-15-10-50 V x (r) OEP1s2s2p3s3p Ar r (a.u.) -40-30-20-100 V x (r) Kr Figure 2: (color online). Orbital–specific exchange potentials V DIMx nl ( r ) and V xOEP , for theground state of He, Ne, Ar and Kr. 10able 3: Orbital and total exchange energies for He, Ne, Ar, and Kr. ❍❍❍❍❍❍ l n nl orbitals have a similar behavior accompanyingthe oep exchange potential. We noticed that the exchange potentials of the orbitals havinga common angular momentum l resemble to each other (see Ar for instance). This wassuggested in a work by Herman et al. where an l –averaged exchange potential for each setof electronic states was calculated as a modification of Slater’s average exchange potential.According to Eq. (10) all orbital–specific potentials should approach the same value at r = 0, since Z DIM nl ( r ) = rV DIM nl ( r ) approaches Z N regardless of nl (the second and third termare the same for every orbital). However, from Fig. 2 it appears that the potentials for thedifferent orbitals approach different values at the origin. This is a consequence of the factthat every DIM potential tends to Z N with different behavior, determined by their fittingparameters. In fact, for very low r values V DIM nl ( r ) ≈ P j α j β j − V d ( r ), but they all havestrictly the same value at r = 0.As a final test for our method, we calculated the total exchange energy E x as given by E x = X nl E x nl = X nl (cid:20) Z ρ HF nl ( r ) V DIMx nl ( r ) dr (cid:21) (11)Table 3 displays the orbital exchange energy as well as the total exchange energy for He, Ne,Ar and Kr. The total exchange energies are compared with the exact atomic Hartree–Fock(EAHF) values given by Becke , with very good agreement. Nitrogen DIM and Exchange Potentials
The procedure developed here is not limited to noble gases or closed shells. As an examplewe will apply the method to Nitrogen. The lower configuration 2 p of Nitrogen gives riseto three different terms: 2 S, 2 D, 2 P. Each of them is described by a different electronicdensity. The fitting parameters that define the term–dependent effective charges are givenin Table 4 for each of the terms. We built the corresponding DIM potentials from theseeffective charges. By using these potentials we solved the Schr¨odinger equation (Eq.(1))for every term, obtained the solutions, the energies, and the corresponding mean values h r i and h /r i . The comparison between the orbitals obtained from the diagonalization of theHamiltonian with the effective potentials and the original Hartree–Fock orbitals are shown11able 4: Fitting parameters for the effective charge Z DIM nl ( r ) for 2 S, 2 D and 2 P terms ofNitrogen. 2 S 2 D 2 P nl α β α β α β s s p h r i obtained with the DIM effective potentials agree with theHF values in about 0 . h /r i mean values agree in about 0 . E DIM for each term of the Nitrogen atom using Eq. (11) are presented in Table5. The agreement between the DIM total energies and the original HF total energies isexcellent, of about 0 . nl –orbital exchange potentials for the 2 S,2 D and 2 P terms, calculated with the depurated inversion method. Again, to compare ourresults, the exchange potential given by Talman ( oep ) is presented in the figures in lightgrey. Figure 3(a) illustrates the exchange potential for the 1 s orbitals for the different terms,showing an overall similarity. The oep potential behaves like the V xDIM1 s ( r ) only at shortand large distances. Figure 3(b) shows the exchange potentials for the 2 s orbitals. In thiscase, noticeable differences between the term–potentials arise at low values of r . For r higherthan 0.5 a.u., all the term–potentials become indistinguishable and agree perfectly with the oep potential. A pecularity observed in the figure is that the oep potential agrees very wellwith the V xDIM2 s ( r ) for the 2 D term. Figure 3(c) displays the 2 p exchange potentials, whichbehave similarly for all the terms. However, since the oep potential is the same for all theorbitals and terms, it disagrees completely with the V x2 p ( r ) at short distances.Table 6 presents the total exchange energy and the nl –exchange energies of the 2 S, 2 Dand 2 P terms. The 1 s –exchange energy for all the terms are the same, as expected for aclosed–shell orbital. Similarly, the 2 s –exchange energy varies slightly, with a difference of0 . p –exchange energy, which varies significantly,having discrepancies of about 18% between the different terms. The total exchange energycomputed with Eq. (11) for the terms are compared with the exact atomic Hartree–Fock(EAHF) exchange energy, with an agreement of about 0 . CONCLUSIONS
A crucial requirement of the density functional method is the accurate representation of theexchange functional. On the other hand, the atomic collision community needs accurate one–electron potentials in order to generate the bound and continuum states on the same footingfor further calculations of collisional processes. These potentials need to be worked out for12able 5: Total and orbital energy and mean values for the 2 S, 2 D and 2 P terms of Nobtained from the DIM effective potentials (upper rows) compared with the Hartree–Fockvalues (lower rows).
E nl ǫ h r i h /r i S − .
376 17 1 s − .
629 06 0 .
228 30 6 .
648 63 − .
400 93 − .
629 06 0 .
228 30 6 .
653 242 s − .
945 32 1 .
334 48 1 .
080 37 − .
945 32 1 .
332 28 1 .
078 182 p − .
567 59 1 .
412 68 0 .
954 98 − .
567 59 1 .
409 63 0 .
957 692 D − .
275 57 1 s − .
666 39 0 .
228 29 6 .
649 29 − .
296 17 − .
666 39 0 .
228 26 6 .
653 882 s − .
963 67 1 .
329 17 1 .
086 44 − .
963 67 1 .
326 32 1 .
083 182 p − .
508 66 1 .
448 78 0 .
938 82 − .
508 66 1 .
446 62 0 .
942 082 P − .
208 56 1 s − .
691 60 0 .
228 24 6 .
650 36 − .
228 10 − .
691 60 0 .
228 24 6 .
654 302 s − .
976 34 1 .
325 62 1 .
087 12 − .
976 34 1 .
322 32 1 .
086 562 p − .
471 30 1 .
471 76 0 .
929 82 − .
471 30 1 .
473 01 0 .
931 55Table 6: Orbital and total exchange energies for 2 S, 2 D and 2 P terms of Nitrogen.1s 2s 2p Total EAHF2 S -2.1175 -0.4776 -0.4711 -6.6034 -6.5962 D -2.1175 -0.4777 -0.4262 -6.46882 P -2.1175 -0.4780 -0.3973 -6.382713 .001 0.01 0.1 1 10 r (a.u.) -7-6-5-4-3-2-10 V x (r) OEP S D P (a) r (a.u.) -7-6-5-4-3-2-10 V x (r) OEP S D P (b) r (a.u.) -6-5-4-3-2-10 V x (r) OEP S D P (c) Figure 3: (color online). DIM exchange potential V DIMx nl ( r ) for the (a) 1 s , (b) 2 s and (c) 2 p orbitals, for the 2 S, 2 D and 2 P terms of Nitrogen.14ny nl –specific orbital, a feature that in general is not present in the chemistry communityfunctionals. In the present work we devised and implemented a depurated inversion method,which allows to obtain the intended potentials through a very simple analytical expressionof the effective charges. The method consists in the inversion of a Kohn–Sham equation,in which the KS orbitals have been replaced by the Hartree–Fock orbitals. By means ofdiagonalization we have achieved accurate wavefunctions having almost perfect agreementwith the original Hartree–Fock wave functions. The quality of the potentials obtained bythe present method is remarkably good. We applied the developed methodology to thecalculation of the ground state orbitals of noble gases and the Nitrogen atom. It is worthmentioning that the same technique can be used for any other level, i.e., it is not limited tothe ground state. ACKNOWLEDGMENTS
This work was supported by grants of CONICET, ANPCyT, and UBACyT, of Argentina.
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