Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment

Transcription

1 Electric ipole an quarupole contributions to valence electron bining in a charge-screening environment A. D. Alhaiari (a) an H. Bahlouli (b) (a) Saui Center for Theoretical Physics, P. O. Box 3741, Jeah 1438, Saui Arabia (b) Physics Department, King Fah University of Petroleum & Minerals, Dhahran 3161, Saui Arabia Abstract: We make a multipole expansion of the atomic/molecular electrostatic charge istribution as seen by the valence electron up to the quarupole term. The Triiagonal Representation Approach (TRA) is use to obtain an exact boun state solution associate with an effective quarupole moment an assuming that the electron-molecule interaction is screene by suborbital electrons. We show that the number of states available for bining the electron is finite forcing an energy jump in its transition to the continuum that coul be etecte experimentally in some favorable settings. We expect that our solution gives an alternative, viable an simple escription of the bining of valence electron(s) in atoms/molecules with electric ipole an quarupole moments. We also ascertain that our moel implies that a pure quarupole-boun anion cannot exist in such a charge-screening environment. PACS: Ge, F, Ca, Ge, p Keywors: electric ipole, electric quarupole, valence electrons, screene 1/r 3 singular potential, triiagonal representations, recursion relation 1. Introuction The interaction of a charge point particle (electron) with extene objects such as large molecules or ions is a problem of funamental interest that receive a lot of attention since the early ays of nuclear an molecular physics [1]. This phenomenon was moele by the interaction of the electron as a point charge with the multipole expansion of the molecular electrostatic potential an may result in its capture in an anion boun state whose electronic orbital is of a iffuse nature. The interaction is ominate by the weak long-range charge multipole attractive potential that scales as a power law of the inverse istance to the center of the molecule. If the ominant potential is ue to the finite ipole moment of the molecule then such a boun state is calle ipole-boun state an playe an important role in electron-molecule interactions an gave rise to valence-boun anions, which were extensively stuie in the literature [1-7]. In recent years the authors obtaine a close form solution of the time-inepenent Schroinger equation for an electron in the fiel of molecule treate as a point electric ipole an euce the critical value(s) of the ipole strength above which bining coul occur for both the groun an excite states [8]. If the molecular ipolar moment vanishes or is negligible while the quarupole moment is ominant, then it is natural to examine the possibility of electron bining by neutral molecules with significant quarupole moments to form quarupole-boun anions in which the long-range charge-quarupole attractive potential ominates [9-1]. The quarupole potential varies as 1/r 3 an is singular for any attractive value of the quarupole moment. Due to the non-central nature of the potential, which oes not allow 1

2 factorization of the electronic wavefunction, exact close form solutions were not possible an only asymptotic solutions were known analytically. One of the first theoretical stuies of potentially quarupole-boun anions was performe by Joran an Liebman [13]. They consiere attachment of an extra electron to BeO imer to form the rhombicbeo cluster, which may be consiere as a quarupoleboun anion. This an similar clusters like MgO [14] an KCl [15] were shown to have relatively high electron bining energies an shoul probably be consiere as potential caniates for quarupole valence-boun anions [9]. However, up to present it has been an experimental challenge to fin systems where the excess electron is boun only ue to the electric quarupole moment [10,16] which resulte in a controversy in the literature. The first experimental claim of a quarupole-boun anion has been obtaine in 004 [17] for the trans-succinonitrile (NC-CH-CH-CN) molecule. The observation of an excite quarupole-boun state for cryogenically coole cyanophenoxie anions has also been reporte recently but it was recognize that the boun state was very weak [18]. Recently, state-of-the-art ab initio calculations on several quarupole-boun anion caniates [19] conclue that the quarupole is much weaker than ipole bining as the electron-molecule potential is not ominate by a single component. Couple cluster abinitio methos were also use recently [0] to stuy electron attachment to covalent molecules an ion clusters with vanishing ipole but large quarupole moments. They foun that the quarupole of the neutral molecular is too weak to bin an excess electron an that there is clearly no such thing as a critical quarupole moment. In this paper, we make a theoretical investigation of the bining of a valence electron in an atomic/molecular system where the charge istribution, as seen by the valence electron, inclues contributions coming from the electric monopole (the net charge), electric ipole, an an (effective) electric quarupole terms. The valence electron interaction with the atom/molecule is assume to be screene by the outer orbital electrons. Moreover, if the net charge (monopole term in the multipole expansion) is zero, then the solution, if it exists, represents the bining of an electron to a neutral molecule with electric ipole an quarupole (i.e., an anion). However, our moel is not solvable for zero net charge. Nonetheless, taking the zero-charge limit results in a zero energy boun state an hence shows that a pure quarupole-boun anion is not allowe in the charge-screening environment of our moel.. Formulation of the problem In the atomic units m 1, the time-inepenent three-imensional Schröinger equation for an electron uner the influence of an electrostatic potential Vr ( ) associate with a given charge istribution reas as follows 1 V ( r ) E ( r ) 0, (1) where is the three-imensional Laplacian, E is the electron energy an ( r ) its wave r r,,, this wave equation becomes function. In spherical coorinates,

3 1 1 1 r 1 y y V( r) E ( r) 0 r r r r y y 1 y, () where y cos. This equation is separable for potentials with the following general form 1 1 V( r) Vr ( r) V ( y) V ( ) r 1 y. (3) 1 If we write the wavefunction as ( r) r R( r) ( ) ( ), then the wave equation () with the potential (3) becomes separate in all three coorinates as follows 1 V ( ) E 0, (4a) 1 E 1 y y V ( y ) E 0 y y 1 y, (4b) 1 r E Vr () r E R 0, (4c) r where E an E are the imensionless angular separation constants. The boun states wavefunction components satisfy the physical bounary conitions: R(0) R( ) 0, ( ) ( ) an (0), ( ) being finite. In References [8] an [1], we obtaine exact solutions for equations (4) with the potential components V ( ) 0, V ( y) Q y an Vr ( r) Qrmaking the total potential function (3) Q cos V( r) Q, (5) r r which is that of an electron interacting with an electric ipole of moment Q line up along the positive z-axis an with a net charge Qe. We took (an will continue to take) the Bohr raius, a0 4 0 me 4 0 e, as the unit of length with m an e being the mass an charge of the electron. For Q 0, the solution obtaine in [8] an [1] refers to an electron boun to a neutral molecule with a permanent electric ipole moment Q (i.e., the ipole-boun anion). In the present stuy, we exten that work by incluing higher orer contributions coming from the electric quarupole. Therefore, we consier an electrostatic potential ue to the istribution of the atomic/molecular charges an in the multipole expansion of the potential, we inclue terms up to the linear electric quarupole where we obtain [] 1 (3cos 1) Q cos V( r) Q p r r r 3, (6) 3

4 where p is the electric quarupole moment. It is obvious that the quarupole term estroys separability of the wave equation an makes the search for an exact solution to this problem a highly non-trivial task. In fact, such exact solution is foun nowhere in the publishe literature. Nonetheless, we make two simplifications to the problem so that we en up with an attainable solution that is still interesting an physically useful. The first simplification is to consier an effective electric quarupole interaction where the angular factor 1 3cos 1 is average over angular space as 1 3 cos 1, where is a real imensionless parameter such that 1 1. This results in an effective 3 quarupole potential Qq r, where the effective electric quarupole moment is Qq p whose value coul be positive or negative. Therefore, the effective potential (6) regains separability where the angular components of the potential remain as V ( ) 0 an 1 3 V ( y) Q y whereas the raial component becomes Vr() r Qr Qqr. Unfortunately, this simplification is not enough to ease our task of fining an exact solution to the effective problem ue to the combination of a highly singular potential in aition to a long-range behavior. Nonetheless, we will show below that by using the TRA we o succee in obtaining an exact solution if we simplify this moel further by making the physically plausible assumption that the electric interaction of the valence electron with the molecule is screene by the outer sub-orbital electron. The size of the screening is ifficult to estimate, however, we can assume that the effective nuclear charge has been reuce. Therefore, the effective net charge istribution as seen by the valence electron inclues the charge of the nucleus Ze an the suborbital inner electrons that are not contributing to the screening, ( Z Qe ), where 1 Q Z. Moreover, the screening is taken in the conventional way resulting in a short-range behavior of the interaction with the valence electron that iminishes exponentially as e r. Consequently, our moel has three free parameters; (i) the screening parameter, (iii) the quarupole parameter, an (iii) the screening charge number Q 1,, 3,.., Z. These three parameters are to be tune an fixe by physical observations for a given atom/molecule. We nee to mention that we are treating our molecular charge istribution as static neglecting its rotational egrees of freeom within the aiabatic limit where its moment of inertia is consiere to be infinite. The exact solution of the angular equations (4a) an (4b) with V ( ) 0 an V ( y) Q y subject to the physical bounary conitions were obtaine in Refs. [8,1]. Intereste reaers may consult the cite work an references therein. Here, we just state results that are relevant to the solution of the raial equation (4c). In [8,1], we obtain E k, where k is the azimuthal quantum number k 0, 1,,... On the other han, the separation constant E belongs to the set of eigenvalues of the following triiagonal symmetric matrix 1 1 nn ( ) ( n1)( n 1) T, n nm 4 nm, Q nm, 1, 1 ( n ) 14 nm ( n 1) 14, (7) In an ieal environment, we can calculate the angular average cos by using the angular wave function ( y), which was obtaine in Ref. [1], as ( y) cos ( y). 4

5 where k 0,1,,... Therefore, for a given value of the electric ipole moment Q, there is an infinite set of values of E for each quantum number. On the other han, for a given, there is a critical value of Q below which the electron cannot stay boune to the neutral molecule ( Q 0 ) to form a ipole-boun anion. That critical ipole moment is the lowest value that makes the eterminant of the matrix T 1 4 I vanish, where I is the unit matrix. This correspons to E 1 4, which is the critical singularity strength of the inverse square potential E r beyon which quantum anomalies appear [3,4]. With this set of values of E for a given, we fin below the exact solution of the raial wave equation (4c) where the raial potential Vr () r behaves near the origin as 1 3 Qr Qr whereas far away it gets iminishe by an electronic screening of the form e r. q 3. The TRA raial solution The exact angular components of the wavefunction, ( ) an ( ), associate with the ipole potential Q cos r are alreay known (see, for example, Ref. [1] an references therein). Hence, to complete the exact solution of the problem, we nee to obtain the raial wavefunction Rr (). We start by writing the raial Schröinger equation E for the effective raial potential of Eq. (4c), Veff () r V () r r, as follows: r V ( ) ( ) 0 eff r ER r, (8) r where we have aopte the atomic units m 1 an took e 4 0. Next, we make a transformation to a imensionless coorinate x() r coth( r), where r 0 an is a real positive scale parameter. Thus, x 1 an Eq. (8) is mappe into the following secon orer ifferential equation in terms of x EVeff ( x) (1 x ) (1 x ) x R r( x) 0, (9) x x (1 x ) In Ref. [5], we use the TRA to obtain an exact series solution of this equation. In the TRA, the solution of the wave equation is written as infinite (or finite) series of square integrable functions, which are require to be complete an to prouce a triiagonal matrix representation for the wave operator [6]. Consequently, the matrix wave equation gives rise to a three-term recursion relation for the expansion coefficients of the series. We solve the recursion in terms of orthogonal polynomials whose properties (e.g., weight function, zeros, asymptotics, etc.) give the physical properties of the system such as the boun states energies, the ensity of states, the scattering phase shift, etc. In Ref. [5], 5

6 the solution of Eq. (9) was obtaine as the series is the set of square integrable basis with elements N n0 n n, where ( x n ) R () r f () x (, ) n( x) cn( x 1) ( x 1) Pn ( x), (10), where Pn( ) ( x) is the Jacobi polynomial, 1, N 1 n 0,1,,..., N an N is a non-negative integer. Therefore, must be negative whereas the normalization sin ( 1) ( n1) ( n1) constant is taken as cn 1 n 1 sin ( n1) ( n1). The TRA compatible potential in Eq. (8) is obtaine in [5] as, B cosh( r) V () eff r Acoth( r) 1 C, (11) 3 sinh ( r) sinh ( r) where A, BC, are real imensionless constants such that A 1 an the basis parameters an are relate to the potential parameter A an the energy as follows, A, (1) where E. Therefore, reality ictates that the energy is negative corresponing to boun states. Now, if we write the expansion coefficients as fn fp 0 n, then P0 1 an (, ) 1 P n becomes the orthogonal polynomial Hn ( z ;, ) efine in [5,7,8] by its symmetric three-term recursion relation cosh H ( z ;, ) D H ( z ;, ) D H ( z ;, ) (, ) 1 (, ) 1 (, ) 1 n n1 n1 n n1 1 (, ) 1 z sinh n FnHn ( z ;, ) 1 (13) where z B C, cosh BC, 1 4, ( n1)( n1)( n1)( n1) n (n1)(n3) F n ( n )(n ) an D n. Thus, reality ictates that BC 1, which is in fact what is neee to prouce a potential configuration that can support boun states. The overall factor f 0 is the square root of the positive efinite weight function of (, ) 1 Hn ( z ;, ). Unfortunately, the analytic properties of this polynomial are not foun in the mathematics literature. This is still an open problem in orthogonal polynomials [8]. Nonetheless, these polynomials coul be written explicitly to all egrees, albeit not in a close form, using the recursion relation (13) an the initial values (, ) H. 1 0 H, (, ) It is obvious that the potential (11) is singular at the origin with r, r an r 3 singularities of strength A, B an C, respectively. However, it has a short range since it r ecays exponentially away from the origin as e. Thus, gives a measure of the extent 6

7 of the range of this potential (i.e., the screening of the electric interaction where the screening parameter ). Near the origin, the potential function (11) behaves as follows V eff 1 A B C () r 3 r r r. (14) On the other han, the effective raial potential of Eq. (4c) in our moel with a net charge Q, electric ipole moment Q, an effective electric quarupole moment Q is q E Q Q E q Veff () r V () r r. (15) 3 r r r r Comparing (14) an (15) suggests that we can use potential (11) to moel the screening of the atomic/molecular charge istribution whose net charge is Q A, effective electric quarupole moment is Qq C an with an electric ipole moment Q whose associate angular contribution is E B. Consequently, the boun state solution obtaine here coul be consiere as an alternative, viable an simple escription of the bining of valence electron(s) in atoms/molecules with electric ipole an quarupole moments. Another conclusion from our finings goes as follows. The short-range behavior ue to screening will permit only a finite number of boun states as will be confirme by the results of our calculation below. Consequently, transition of the valence electron to the continuum will not be smooth but a jump is expecte ue to the finite energy ifference between the highest excite state an the continuum. In a favorable experimental setting, it might be possible to etect this energy jump. However, in the absence of screening ( 0 ), the long-range behavior of the interaction allows for an infinite number of boun states with infinitesimally small spacing an ensely packe at the top of the energy spectrum near the continuum. This makes transition of the valence electron to the continuum smooth. Moreover, since boun state solution requires BC1, then this implies that E Qq an since E epens on the azimuthal quantum number, then this puts a boun on these quantum numbers ( 0,1,,.., max ) that epens on the effective quarupole moment Q q. Aitionally, since A 1 an A Q, then for a given charge Q the value of the screening parameter must be chosen from within the range 0 4Q. We coul, in principle, use our moel as an extension to, an/or generalization of, the celebrate ipole-boun anion problem if we coul solve it for Q 0. However, as seen above the solution requires that the net charge of the atom/molecule be non-zero an positive (i.e., ionize molecule) since A 1 an Q A. Nonetheless, in the limit of extremely week screening (i.e., 0) a quaruple-boun anion is, in principle, possible since the electric charge on the molecule tens to zero but we must also let C 0 such that the ratio C Q q. However, the resulting bining energy, which is E, will ten to zero implying that the electron cannot stay boun to form a stable quarupole-boun anion. Therefore, we conclue that a pure quarupole-boun anion cannot exist in such an environment. These finings are consistent with the experimental 7

8 observations that there has been no firm evience for the existence of weakly boun anions ue to the quarupolar interactions alone [17]. 4. Numerical Proceure In the absence of knowlege of the analytic properties of the orthogonal polynomial (, ) 1 Hn ( z ;, ), we resort to numerical techniques to obtain the energy spectrum. Due to the energy epenence of the basis elements via the parameters an as shown by Eq. (1), we may use one of two numerical proceure (i) the potential parameter spectrum (PPS) techniques or (ii) the irect iagonalization of the Hamiltonian matrix (HMD) in an energy inepenent basis similar to those in (10). In the latter, we use Gauss quarature integral approximation associate with the Jacobi polynomials. We will not give etails of these two techniques. Intereste reaers may consult Refs. [9,30] for a escription of the PPS technique. In the Appenix, we give a brief outline of how to obtain the matrix elements of the Hamiltonian an compute the energy spectrum using Gauss quarature integral approximation. To evelop our moel for a given atom/molecule with an atomic/molecular charge number Z, an electric ipole moment Q an an electric quarupole moment p, we follow a proceure with the following steps: 1. Construct a large enough size Table for the eigenvalues E of the triiagonal matrix T in (7) with the azimuthal quantum number 0,1,,... For each label the sorte eigenvalues E by an integer m 0,1,,... Therefore, a pair of the azimuthal quantum numbers an m uniquely ientifies E.. Choose a value for the net charge (monopole term) Q from the set 1,,.., Z. 3. Choose a value for the charge screening parameter from within the range 0 4Q. 4. Choose a value for the electric quarupole parameter from within the range Impose the constraint E p on the Table obtaine in step 1 above. The result is a sub table for E whose columns are inexe by 0,1,,.., max with th each column having a finite number of rows m0,1,,.., m. 6. Take the following potential parameters: A Q, B E an C p, where E is one of the values from the sub table in the previous step labele by the pair of azimuthal quantum numbers an m. 7. With the potential parameters A, BC, from the previous step, use one of the two methos, PPS or HMD, to compute the bining energy of the valence electron to the atom/molecule an compare to physical observations. Repeat the calculation for all values of E from the sub table in step 5 above. 8. If an agreement with experimental observations is not reache then go back to step above to tune the parameters an repeat the calculation with an alternate set Q,,. 8

9 9. Once an agreement is reache then register the set of moel parameters, an quantum numbers Q,, m that will be use to escribe the bining of the valence electron to the atom/molecule whose atomic/molecular charge number Z, electric ipole moment Q an electric quarupole moment p. As an example, we consier a hypothetical molecule with charge number Z 1, electric ipole moment Q 35 an electric quarupole moment p 7. Table 1 gives the energy spectrum of the valence electron for a moel parameters Q,, 8,0. Q,0.3 an for the liste azimuthal quantum numbers,m corresponing to the shown values of E. The Table shows that favorable settings to measure the energy jump to the continuum are obtaine if the valence electron is in a state corresponing to the azimuthal quantum numbers,m (0,), (1,1) an (,0). 5. Conclusion In this work, we have focuse on how to escribe the weak bining of an external electron to a stable positively charge molecule using a single electron Schroinger equation with an effective interaction potential that involves the attractive Coulomb interaction at large istances in aition to higher orer ipole an quarupole of a static molecular charge istribution. We have use the triiagonal representation approach to fin the bining energy of the valence electron to an atom/molecule with an electric ipole an quarupole but with charge screening coming from suborbital electrons. We showe (as it is always the case for the screene Coulomb interaction) that the number of states available for bining the electron is finite, which might give rise to an energy jump in its transition to the continuum that coul be etecte experimentally uner some favorable conitions. The moel we presente is not solvable for zero net charge an hence cannot escribe an anion. However, taking the limit of zero net charge in the moel results in a vanishing bining energy, which means that our moel asserts that a quarupole-boun anion is not allowe in a screening environment. These finings are consistent with experimental observations that there has been no clear evience for the existence of weakly boun anions ue to the quarupolar interactions alone [17]. Our present moel is mainly influence by the asymptotic inverse cubic behavior of the quarupolar molecular potential an neglects etails relate to the short-range nature of the potential relate to the anisotropy of the molecular charge istribution an i not take into consieration the rotational motion of the molecule. Regarless of the above shortcomings, we expect that our solution gives an alternative, viable an simple escription of the bining of valence electron(s) to static positively charge atoms/molecules having finite electric ipole an quarupole moments. 9

10 Appenix: Hamiltonian matrix, Gauss quarature an evaluation of the energy spectrum The matrix elements of the Hamiltonian operator in the energy inepenent basis elements (10) is obtaine in Ref. [5] as H ( x) H ( x) r m n 0 m n (, ) (, ) cc m n ( x 1) ( x 1) WxP ( ) ( ) ( ) 1 m xpn xx (A1) A 1 where ( ) W x n BCx an we use the integral 1x 1x 4 transform x r 0 1 x. The recursion relation an orthogonality of the Jacobi 1 k polynomial show that all terms in W(x) that are proportional to x, where k is a nonnegative integer, will prouce exact matrix elements that are k 1 iagonal (i.e., a bane iagonal matrix whose iagonal ban is of with k 1). Therefore, only the first two terms in W(x) will prouce a matrix with non-zero entries everywhere, which nees to be evaluate numerically. We can eliminate one or both of these terms by choosing 0 an/or A. However, if we eliminate both then the conition N 1 will ictate that N 1 A 1, which restricts severely the size of the matrices an hiner the accuracy of the calculation. Therefore, we eliminate only the first term, which is singular at x 1, an choose 0 while keeping arbitrary but such that N 1. To calculate the matrix elements that represent the non-zero secon term in W(x), we use Gauss quarature integral approximation associate with the Jacobi polynomial. The etails of this metho are foun in Refs. [31,3]. The essence of the metho might be summarize as follows. Let X nm, be the matrix elements obtaine by choosing W( x) x in Eq. (A1). Then by using the recursion relation an orthogonality of the Jacobi polynomial it is easy to show that X reuces to the following triiagonal symmetric matrix X F D D, (A) nm, n nm, n nm, 1 n1 nm, 1 where the numbers Fn, D n are efine in section 3 below Eq. (13) (with 0 ). Now, N let nn 0 be the eigenvalues of the ( N 1) ( N 1) truncate version of the matrix X N an mn, m be its normalize eigenvector associate with the eigenvalue 0 n. Then an approximate evaluation of the matrix elements that represent any function W(x) is W T nm,, (A3) nm, 10

11 where is a iagonal matrix with elements nm, W ( n) nm,. Therefore, the evaluate Hamiltonian matrix has finally the following elements where 1 A T H nm, n 1 4B1nm, CXnm, nm, 4 nm, nm,. The size of this matrix (number of basis elements) is 1 1 n, (A4) N with nm, 0,1,,..., N an N 1. To calculate the energy spectrum, we write the energy eigenvalue equation (wave equation) H E in the energy inepenent basis as fmh m E fm m, we obtain m. Projecting from left by m n Hnm, f m m E m nm, fm, where is the basis overlap matrix whose elements are nm, n m. Consequently, the energy spectrum is calculate as the generalize eigenvalues of the matrix equation H f E f. To obtain the overlap matrix, we follow the same Gauss quarature proceure outline above giving T nm,, (A5) nm, nm, where nm,. Finally, we vary the value of until a plateau of computational 1 n stability of the energy spectrum is reache for large enough N. It turns out that the plateau of computational stability (range of values of with no significant change in the result) is larger for lower boun states. 11

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14 Table Caption Table 1: The energy spectrum of the valence electron (in units of 1 ) for a hypothetical molecule with ZQ,, p 1,35,7 in the atomic units. The physical parameters are taken as Q,, 8,0. Q,0.3 so only Q 1 7 outmost suborbital electrons are contributing to the screening. The azimuthal quantum numbers (, m) an the corresponing value of E are shown an we have use the HMD metho outline in the Appenix with a basis size of 100. For the isplaye accuracy, the plateau of N 1, N 1 with N 100. computational stability was taken as Table 1 (, m) E (0,0) (0,1) (0,) (1,0) (1,1) (,0) (,1) (3,0) (4,0)