Analysis of Compton profile through information theory in H-like atoms inside impenetrable sphere
aa r X i v : . [ qu a n t - ph ] F e b Analysis of Compton profile through information theory in H-like atoms insideimpenetrable sphere
Neetik Mukherjee ∗ and Amlan K. Roy † Department of Chemical Sciences Indian Institute of Science Educationand Research (IISER) Kolkata, Mohanpur-741246, Nadia, WB, India
Abstract
Confinement of atoms inside various cavities has been studied for nearly eight decades. However, theCompton profile for such systems has not yet been investigated. Here we construct the Compton profile(CP) for a H atom radially confined inside a hard spherical enclosure, as well as in free condition . Some exactanalytical relations for the CP’s of circular or nodeless states of free atom is presented. By means of a scalingidea, this has been further extended to the study of an H-like atom trapped inside an impenetrable cavity.The accuracy of these constructed CP has been confirmed by computing various momentum moments. Apartfrom that, several information theoretical measures, like Shannon entropy ( S ) and Onicescu energy ( E ) havebeen exploited to characterize these profiles. Exact closed form expressions are derived for S and E usingthe ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits therate of dissipation of kinetic energy. At a fixed ℓ , this rate diminishes with rise in n . However, at a certain n , this rate accelerates with progress in ℓ . A similar analysis on the respective free counterpart displays anexactly opposite trend as that in confined system. However, in both free and confined environments, CPgenerally gets broadened with rise in Z . Representative calculations are done numerically for low-lying statesof the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurateeigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined Hatom (and isoelectronic series) and investigated adopting an information-theoretic framework. PACS:
Keywords:
Compton effect, quantum confinement, H-like atom, information theory, Shannon entropy,Onicescu energy ∗ Email: [email protected]. † Corresponding author. Email: [email protected], [email protected]. . INTRODUCTION In quantum mechanics, electron density (ED) is the most important physical quantity and liesin the heart of chemistry. Because, it is inherently connected with the structure, bonding andreactions-the fundamental pillars. According to Hohenberg-Kohn theorem of density functionaltheory, when the exact density is known, any property of the system can be calculated exactly.It is larger near the nuclei and covalent chemical bonds. All kinds of chemical interactions affectED. X-ray diffraction has provided the experimental information about ED. Similarly, electronmomentum density (EMD) which is the momentum counterpart of ED, has also been used quiteextensively to understand chemical systems. It can be extracted directly from ( e, e ) spectroscopy,position annihilation spectroscopy or X-ray Compton scattering (CS) [1, 2].Compton effect is an inelastic scattering process [1], with use as spectroscopic probe in bothsingle-particle excitation as well as in collective mode [3]. Linear CS for weekly bound electronsis well explained by the so-called impulse approximation [4], where the electron is assumed tobe quasi-free. It means that, the binding energy of the electron is insignificant compared to theenergy acquainted to it by the photon, so that the ultimate state of an electron may be adequatelyrepresented by a plane wave [5]. Within such an approximation, one can write,d σ dΩd ǫ = C ( ǫ , ǫ , φ ) J ( q ) . (1)This equation indicates that, double differential cross section ( σ ) measures the quantity of photonscattered by matter having a solid angle Ω with energy ǫ . Here, ǫ signifies the energy of incidentphoton and φ the scattering angle. C ( ǫ , ǫ , φ ) depends on the experimental setup, while J ( q )corresponds to the Compton profile (CP). It determines the projection of EMD along directionof scattering vector. Importantly it provides the intensity of Compton band. Finally, q is theprojection of target electron momentum upon scattering vector. The concept of CS is well known;for its useful properties and features, one can consult some of the excellent elegant reviews [4, 6, 7].CS provides information about EMD. They can then be employed to estimate various momen-tum moments, h p n i . In atomic systems CP was utilized to compute momentum moments, whichare directly connected to entropy optimization principle [8, 9]. It was also invoked to analyzeisoelectronic atoms [8]. In last two decades, numerous attempts were made to adopt CPs in inter-preting various physical and chemical properties in atoms, molecules and solids, both theoreticallyas well as experimentally [10]. The cutting-edge X-ray CS technique permits us to visualize thebonding in liquids [11] and imaging the hole states of dopants in complex materials [12]. Theoreti-2al prediction of CP has also been a very fruitful and worthwhile research topic ever since the workof [13]. The agreement between theory and experiment was improved by incorporating electroncorrelation in the wave function progressively accurately. Substantial amount of work has beenreported using wave-function based techniques as well as density functional theory (DFT). A largeset of closed-shell molecules were studied at Hartree-Fock (HF), various post-HF and DFT levels[14]. Role of basis set in this context is very critical and has been examined [15]. Several suchcalculations adopting CI with singles excitation [16], singles-doubles excitations [17] and multipleexcitations (up to six fold) [18] were reported. Further, CI calculation by perturbation with multi-configurational zeroth-order wave function via iterative processes has been presented [19]. In DFT,the accuracy of CPs strongly depends on quality of exchange-correlation functional[20].Many important concepts in physics such as electron correlation [21, 22], EMD [23–25], Fermisurface determination [26–29], X-ray and γ -ray radiations [30–32], were probed through the helpof CP. CP is successfully employed to understand the anisotropy in nature of hydrogen bondof crystalline ice [33]. Similarly, the hydrogen bond signature in NH F has been studied [34].Interestingly, it can explain the metal-insulator transition in La (2 − x ) Sr (1+2 x ) Mn O [35]. However,in spite of such wide range of applications, CP in confined quantum systems has been rarelyinvestigated neither experimentally nor by theoretical methods (pertaining a few examples), whichwe attempt to explore in this work.Ever since its inception, confinement of quantum system has emerged as a subject of topicalinterest in the field of physics, chemistry, biology, nano-science and nano-technology, attractinga large number of elegant books and review articles [36–43]. Atoms, molecules constricted undercavities of varying size and shape, exhibit distinct fascinating changes in their physical and chemicalproperties from their free counterpart. Very recently, a new virial-like theorem has been proposedfor these systems [44]. Extreme high pressure (of the order of multi-megabar) always influencesalmost all properties of a chemical system, including (i) the fate of a chemical reaction, (ii) reductionin size of anion (iii) elongation of length in covalent bond (iv) increase in coordination number ofan atom in a coordination complex, etc. At such high pressure range, new bond can be formed andexisting ones gets deformed (usually shortened, but in certain cases, stretched too) [38]. Besides,the van der Waal’s space gets compressed [46]. Atom under high pressure was first studied asearly as in 1937 [45]. Such a situation can be modeled by shifting the spatial boundary frominfinity to a certain finite region. Depending upon the capacity of pressure one can simulatethem by invoking two broad category of confining potentials, impenetrable (hard) and penetrable(soft) . The effect of pressure on CP as well as on autocorrelation functions of MgO polymorphs3ere studied in the framework of DFT employing periodic linear combination of atomic orbitalmethod [47]. Interestingly, the experimental investigation of CS under high pressure was donebefore, for elemental silicon by utilizing synchrotron radiation and Mao-Bell version of the Merrill-Basset diamond anvil cell with a Be gasket up to a pressure of 20 Gpa. Moreover, the use of Lauemonochromator and a special assembly of compound refractive lenses made this novel experimentalsetup successful. The detailed description about such unique establishment is available in [48].In the last twenty years, emergence of information theoretical concept has provided a majorimpetus in many diverse field of science and technology [49]. They characterize the single-particledensity of a system (in conjugate r and p spaces) in different complementary ways. Arguably,these are the most eligible measures of uncertainty, as they do not make any reference to somegiven point of a corresponding Hilbert space [50]. Moreover, these are compactly connected toenergetics and experimentally measurable quantities of a given system. Shannon entropy ( S ) isthe arithmetic mean of uncertainty. Onicescu energy ( E ) is the expectation value of density andit is also termed as second-order moment of density [51]. E is also called dis-equilibrium , as itmeasures the deviation of a distribution from equilibrium [52]. Importantly, being reciprocallyconnected to S , E usually upholds the inferences obtained from S . Particularly, in a given space,the increase of spreading in a density distribution is quantified by an increment in S and decayin E . Both S, E are successfully employed to quantify various density-distributions producedfrom relevant theoretical or experimental processes. S has connection to Colin conjecture [53, 54],atomic avoided crossing [55], electron correlation effect [54], configuration interaction [56], quantumentanglement in artificial atom [57, 58], bond formation [59], elementary chemical reactions [60],orbital-free DFT [61], aromaticity [62] etc. Some of these like the avoided crossing occurs underconfinement condition. Further, S has its distinct ability to characterize artificial quantum systemsdesigned by placing an atom or molecule inside a foreign environment [40]. Likewise, E has beenwidely employed to investigate the correlation energy and first ionisation potential in atomic andmolecular systems [63]. In stressed condition, the change in confinement strength leads to thevariation in electron density distributions. This effect can be characterized by employing theseinformation measures.In this article, we intend to show the effect of high pressure on CP of a confined quantum systemconsidering a H-like atom under a rigid impenetrable well (solid hydrogen), as test case. The firstprinciples study of an atom trapped inside a fullerene cavity can provide accurate results. But theymay not direct us to a simple interpretation of the calculated properties [64]. The cavity modelhas been designed using experimental results [65]. In the present scenario, the motive is to extract4 qualitative idea about the impression of high pressure on CPs of CHA. To a certain extent,this can serve the purpose. The designed CPs are characterized by employing a few information-theoretic measures, which helps uncover the effect of confinement on CP. These measures will actas descriptor in interpreting the CPs. This investigation will act as a threshold to discern theinfluence of various confined environments (such as quantum dot, encapsulated atom in fullerenecavity [65] and so on) on CPs. It is worthwhile noting that, CP for a free H atom (FHA) wasstudied before by some researchers [5–7, 13], however, we are not aware of any similar work in aconfined H atom (CHA). Besides, the information analysis of CP in either FHA or CHA is notreported as yet. Therefore, at first, we examine the CP in an FHA; it is found to be possibleto obtain a generalized expression for circular ( n − l = 1) states, while for other states it needsto be calculated numerically. In the next step, we analyze the Compton Shannon entropy ( S c ),Compton Onicescu energy ( E c ) and generalized entropic moment ( α order) using the Comptondensity, in various states in FHA. As it turns out, these quantities can be derived in closed form only for the lowest state of FHA. Now to pursue the above relevant quantities in a CHA, we applythe following form of potential: v c = ∞ at r ≥ r c , and zero elsewhere. Here, v c represents theperturbing potential and r c the confining radius. The calculated CPs were tested by evaluatingseveral momentum moments. It is well established that one can calculate the expectation valueof any of the desired momentum moments in a given state, starting from their respective CP [5].Later, this entire idea has been envisaged to confined H-like atoms (with varying Z , the nuclearcharge) by introducing a novel scaling relation. For our calculations, we have chosen six statescorresponding to n ≤
3, which suffices the current purpose. These are done by means of two wavefunctions, viz , exact wave function available in terms of Kummer hypergeometric function and theaccurate wave function obtained through a generalized pseudospectral (GPS) method. It is verifiedthat the results of these two methods are practically identical. The article is organized as follows.Section II presents a brief summary of various aspects of methodology used in this work. Section IIIprovides an in-depth discussion of the results for both FHA and CHA. Finally we conclude with afew remarks and future prospects, in Sec. IV. II. METHODOLOGYA. Theoretical formalism (cid:20) − d dr + l ( l + 1)2 r + v ( r ) + v c ( r ) (cid:21) ψ n,l ( r ) = E n,l ψ n,l ( r ) , (2)where v ( r ) = − Z/r ( Z = 1 for H atom). Our desired high pressure confinement is established byinvoking the potential: v c ( r ) = + ∞ for r > r c , and 0 for r ≤ r c , where r c implies the radius of thebox. This equation needs to be solved under Dirichlet boundary condition, ψ n,l (0) = ψ n,l ( r c ) = 0.The exact wave function for CHA is obtained by solving Eq. (2), which is expressible in terms ofKummer’s M function (confluent hypergeometric) [66], ψ n,ℓ ( r ) = N n,ℓ (cid:16) r p − E n,ℓ (cid:17) ℓ F " ℓ + 1 − p − E n,ℓ ! , (2 ℓ + 2) , r p − E n,ℓ e − r √ − E n,ℓ , (3)with N n,ℓ denoting normalization constant and E n,ℓ corresponding to energy of a state representedby n, ℓ quantum numbers. At r = r c , this equation becomes zero . It is a transcendental typeequation and becomes useful when E n,ℓ are known. At r = r c , F " ℓ + 1 − p − E n,ℓ ! , (2 ℓ + 2) , r c p − E n,ℓ = 0 . (4)For a certain ℓ , first root confirms the energy of the lowest- n state ( n lowest = ℓ + 1), withsuccessive roots signifying excited states. It is instructive to mention that, to construct the exactwave function of CHA for a definite state, one needs to provide energy eigenvalue of that state.In current purpose, E n,ℓ are computed by means of the GPS method, which has produced highlyaccurate eigenvalues for a number of central potentials, in both free and confined condition [67–69].In this present communication, our objective is to construct the spherically averaged EMD,which is the foundation to generate the CPs. In order to do that, at first, the p -space wavefunction can be obtained numerically by the following standard equation, ψ n,ℓ ( p ) = 1(2 π ) Z ∞ Z π Z π ψ n,ℓ ( r ) Θ( θ )Φ( φ ) e ipr cos θ r sin θ d r d θ d φ, = 12 π r ℓ + 12 Z ∞ Z π ψ n,ℓ ( r ) P ℓ (cos θ ) e ipr cos θ r sin θ d r d θ. (5)Note that, ψ ( p ) needs to be normalized. Integrating over θ and φ variables, this equation can befurther modified to, ψ n,ℓ ( p ) = ( − i ) ℓ Z ∞ ψ n,ℓ ( r ) p f ( r, p )d r. (6)6epending on ℓ , this can be expressed in following simplified form ( m ′ starts with 0), f ( r, p ) = m ′ < ℓ X k =2 m ′ +1 a k cos prp k r k − + m ′ = ℓ X j =2 m ′ b j sin prp j r j − , for even ℓ,f ( r, p ) = m ′ = ℓ − X k =2 m ′ a k cos prp k r k − + m ′ = ℓ − X j =2 m ′ +1 b j sin prp j r j − , for odd ℓ. (7)The coefficients a k , b j of even- ℓ and odd- ℓ states are obtained analytically. In order to get furtherdetails, please see the Tables I and II in reference [43].The spherically averaged EMD or the mean radial distribution function for a definite n, ℓ -statein p space, I n,ℓ ( p ), can be extracted as [5, 8], I n,ℓ ( p )d p = Z ω (cid:2) ψ n,ℓ ( p ) ψ ∗ n,ℓ ( p ) p (cid:3) d p = ψ n,ℓ ( p ) ψ ∗ n,ℓ ( p ) p d p, (8)where ω is an element of solid angle for p . In a given state, I n,ℓ ( p )d p signifies the probability that p has a magnitude between p and p + dp . Therefore, R ∞ I n,ℓ ( p )d p = 1. Now, assuming the impulseapproximation, the desired spherically averaged CP of a CHA can be expressed as follows, J n,ℓ ( q ) = Z ∞| q | I n,ℓ ( p ) p d p. (9) J n,ℓ ( q ) is a function merely of q , with its peak at q = 0, while q is the projection of target-electronmomentum upon the scattering vector. The momentum moments are then given by, (cid:28) p (cid:29) n,ℓ = 2 J n,ℓ ( q = 0) , h p m i n,ℓ = 2( m + 1) Z ∞ q m J n,ℓ ( q )d q. (10)Here, our interest is to estimate S, E for CPs; these can be easily expressed in terms of J ( q ) as, S cn,ℓ = − Z J n,ℓ ( q ) ln J n,ℓ ( q )d q, E cn,ℓ = Z J n,ℓ ( q )d q. (11)A deeply bound electron has a very flat and broad momentum distribution [4]. As a consequence,its CP is also broad. This broadness of distribution can be quantified by S c , E c . Hence, thesemeasures can act as descriptor about the bound effect on an electron within a quantum system.Our current study will convincingly establish this interpretation. B. CHA isoelectronic series
In case of the isoelectronic series of CHA, it is interesting to investigate the influence of Z on CPs. In this sub-section, analytical relations between CP and Z have been established by7sing some scaling properties. The required radial Schr¨odinger equation, in terms of a constrainedcoulomb potential, can be written as, − ¯ h m ∇ ψ n,ℓ ( r ) − Zr ψ n,ℓ ( r ) + V θ ( r − r c ) ψ n,ℓ ( r ) = E n,ℓ ψ n,ℓ ( r ) ,θ ( r − r c ) = 0 at r ≤ r c , θ ( r − r c ) = 1 at r > r c . (12)Here θ ( r − r c ) is the Heaviside theta function and V is taken to be an infinitely large positivenumber. After doing some straightforward mathematical manipulations, by transforming r = λr and assuming λ = ¯ h mZ one leads to the following form of Hamiltonian, − ∇ ψ n,ℓ ( r ) − r ψ n,ℓ ( r ) + V θ (cid:18) r − mZr c ¯ h (cid:19) ψ n,ℓ ( r ) = ¯ h mZ E n,ℓ ψ n,ℓ ( r ) . (13)where r is the new variable with dimension M − L − T I. We assume ¯ h mZ V ≈ V as V approachesto ∞ . Now, a comparison of Eqs. (12) and (13) yields, E n,ℓ (cid:18) ¯ h m , Z, r c (cid:19) = mZ ¯ h E n,ℓ (cid:18) , , mZr c ¯ h (cid:19) ,ψ n,ℓ (cid:18) ¯ h m , Z, r c , r (cid:19) = Aψ n,ℓ (cid:18) , , mZr c ¯ h , r (cid:19) . (14)Applying the normalization condition, we obtain A = λ − . The p -space counterpart of ψ n,l can beachieved by performing the Fourier transformation. It is important to mention that, p = p λ . φ n,ℓ (cid:18) ¯ h m , Z, r c , p (cid:19) = λ φ n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) . (15)Therefore, the spherically averaged momentum density ( I n,ℓ ) will take the form, I n,ℓ (cid:18) ¯ h m , Z, r c , p (cid:19) p d p = φ n,ℓ (cid:18) ¯ h m , Z, r c , p (cid:19) φ ∗ n,ℓ (cid:18) ¯ h m , Z, r c , p (cid:19) p d p = λ φ n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) φ ∗ n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) (cid:16) p λ (cid:17) d p λ = I n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) p d p . (16)Some further manipulation leads to the expression, h p m i n,ℓ = Z p m I n,ℓ (cid:18) ¯ h m , Z, r c , p (cid:19) p d p = Z (cid:16) p λ (cid:17) m I n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) p d p = 1 λ m Z p m I n,ℓ (cid:18) , , mZr c ¯ h , p (cid:19) p d p = h p m i n,ℓ λ m . (17)Our desired spherically averaged Compton density can be achieved adopting the following strategy, J n,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = Z ∞| q | I n,ℓ (cid:16) ¯ h m , Z, r c , p (cid:17) p p d p. (18)8ere, p = p λ , then | q | = | q | λ and d p = d p . Hence, applying Eq. (16), J n,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = λ Z ∞ | q | λ I n,ℓ (cid:16) , , mZr c ¯ h , p (cid:17) p p d p = λ J n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) . (19)The above equation finally corresponds to the CP for CHA isoelectronic systems. Now, let us tryto evaluate various momentum moments as below, h p m i n,ℓ = 2( m + 1) Z ∞ q m J n,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) d q = 2( m + 1) Z ∞ (cid:16) q λ (cid:17) m λJ n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) d q λ = 1 λ m m + 1) Z ∞ q m J n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) d q = h p m i n,ℓ λ m . (20)Thus Eqs. (17) and (20) have provided exactly identical expression, implying that our scalingformula in constructing CP for isoelectronic series is correct. Following a similar argument, onecan easily write, D p E n,ℓ = λ D p E n,ℓ .Now, using Eq. (11) and Eq. (19) one can get, S cn,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = − Z J n,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) ln J n,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) d q = − Z λJ n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) (cid:18) ln λ + ln J n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19)(cid:19) d q λ . (21)Since R J n,ℓ (cid:16) ¯ h m , Z, r c , q (cid:17) d q = R J n,ℓ (cid:16) , , mZr c ¯ h , q (cid:17) d q = , we can write, S cn,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = −
12 ln λ − Z J n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) ln J n,ℓ (cid:18) , , mZr c ¯ h , q (cid:19) d q S cn,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = −
12 ln λ + S n,ℓc (cid:18) , , mZr c ¯ h , q (cid:19) . (22)An analogous mathematical exercise will lead us to the following expression for Onicescu energy, E cn,ℓ (cid:18) ¯ h m , Z, r c , q (cid:19) = λE n,ℓc (cid:18) , , mZr c ¯ h , q (cid:19) . (23)As mentioned earlier, the present calculations are done by engaging two different wave functions,namely, (i) exact wave function given in Eq. (3) and (ii) those obtained numerically through theGPS scheme. In either case, Compton profiles are established numerically in p space. The tworesults are found to compliment each other in all occasions. Henceforth, we will report resultsproduced from the exact wave function only. 9 II. RESULT AND DISCUSSION
The results will be discussed in three subsections. At first we present the CP and Comptoninformation quantities on ground and some low-lying excited states of FHA. This offers an oppor-tunity to derive the CPs in nodeless states in closed analytical form. This will help us to distinguishthe manifestation of CP and all other measures from FHA to CHA, which is taken up in the secondstage. Thus we will discuss and characterize the freshly built CPs of H-atom under high pressureenvironment, in detail. Finally, the above analysis will be extended to isoelectronic series of CHA.
A. Free H-like atom
In case of FHA like systems ( r c → ∞ ) the first-order hypergeometric function modifies toassociated Laguerre polynomial with E n = − Z n . Therefore, Eq. (3) reduces to, ψ n,ℓ ( r ) = 2 n (cid:20) ( n − ℓ − n + ℓ )! (cid:21) (cid:20) Zn r (cid:21) ℓ e − Zn r L (2 ℓ +1)( n − ℓ − (cid:18) Zn r (cid:19) . (24)It is well-known that the Schr¨odinger equation of H-like atom in p space can be solved exactly.The exact p -space wave function [70], in this case, assumes the following form, ψ n,ℓ ( p ) = n Z (cid:20) π ( n − ℓ − n + ℓ )! (cid:21) (2 ℓ +2) ℓ ! n ℓ { [ npZ ] + 1 } ℓ +2 (cid:16) pZ (cid:17) ℓ C ℓ +1 n − ℓ − (cid:18) [ npZ ] − npZ ] + 1 (cid:19) , (25)where C ηζ ( t ) signifies the Gegenbauer polynomial. In what follows, we attempt to provide anaccurate analytical form of CP for nodeless states, by designing a generalized equation. From this,simplifying expressions are given for first five circular or nodeless states. For non-circular states,they become somehow involved and we have not pursued as it is aside the main objective of thiswork. On the other hand, the exact forms of S cn,ℓ and E cn,ℓ could be derived only for the groundstate; for all excited states, recourse has been taken to numerical method.We note that, in a circular state ( n − ℓ ) = 1 and C ℓ +1 n − ℓ − (cid:16) [ npZ ] − npZ ] +1 (cid:17) reduces to unity. Thus, theradial component in p space simplifies to, ψ n − ℓ =1 ( p ) = n Z (cid:20) π n + ℓ )! (cid:21) (2 ℓ +2) ℓ ! 1 { [ npZ ] + 1 } ℓ +2 (cid:16) npZ (cid:17) ℓ . (26)Therefore, the spherically averaged EMD or radial density in p space takes the form, I n − ℓ =1 ( p ) = n Z (cid:20) π n + ℓ )! (cid:21) (4 ℓ +4) ( ℓ !) { [ npZ ] + 1 } ℓ +4 (cid:16) npZ (cid:17) ℓ . (27)Now, using the definition in Eq. (9) and employing Eq. (27), the CP can be expressed as, J n − ℓ =1 ( q ) = 12 n Z (cid:20) π n + ℓ )! (cid:21) (4 ℓ +4) ( ℓ !) Z ∞| q | { [ npZ ] + 1 } ℓ +4 (cid:16) npZ (cid:17) ℓ p d p. (28)10y setting (cid:0) npZ (cid:1) = y , Eq. (28) can be rewritten as, J n − ℓ =1 ( q ) = n Z (cid:20) π n + ℓ )! (cid:21) (4 ℓ +4) ( ℓ !) Z ∞ ( nqZ ) y ℓ ( y + 1) ℓ +4 d y. (29)Now adopting the binomial expression and replacing y ℓ = ( y +1) ℓ − P ℓk =1 ℓ ! k !( ℓ − k )! y ( ℓ − k ) , in Eq. (29),one arrives at the following generalized form, J n − ℓ =1 ( q ) = n Z (cid:20) π n + ℓ )! (cid:21) (4 ℓ +4) ( ℓ !) Z ∞ ( nqZ ) [( y + 1) ℓ − P ℓk =1 ℓ ! k !( ℓ − k )! y ( ℓ − k ) ]( y + 1) ℓ +4 d y. (30)This is our general expression for J n − l ( q ) in circular states of H-like atoms. Using appropriatevalues of n, ℓ in Eq. (30), following expressions can be written for five such lowest states: J s ( q ) = 83 πZ h(cid:0) qZ (cid:1) + 1 i , (31) J p ( q ) = 6415 πZ (cid:20) (cid:16) qZ (cid:17) + 1 (cid:21)(cid:20)(cid:16) qZ (cid:17) + 1 (cid:21) , (32) J d ( q ) = 3072525 πZ (cid:20) (cid:16) qZ (cid:17) + 7 (cid:16) qZ (cid:17) + 1 (cid:21)(cid:20)(cid:16) qZ (cid:17) + 1 (cid:21) , (33) J f ( q ) = 163842205 πZ (cid:20) (cid:16) qZ (cid:17) + 36 (cid:16) qZ (cid:17) + 9 (cid:16) qZ (cid:17) + 1 (cid:21)(cid:20)(cid:16) qZ (cid:17) + 1 (cid:21) , (34) J g ( q ) = 13107214553 πZ (cid:20) (cid:16) qZ (cid:17) + 165 (cid:16) qZ (cid:17) + 55 (cid:16) qZ (cid:17) + 11 (cid:16) qZ (cid:17) + 1 (cid:21)(cid:20)(cid:16) qZ (cid:17) + 1 (cid:21) . (35)It is interesting to mention that the expressions for 1 s, p states in Eq. (31) and (32) are the sameas those given long times ago in [13]. However the generalized expression, Eq. (30) apparently hasnot been reported before. Similar exercise can also be undertaken for non-circular states as well.But these are cumbersome and outside the main focus of this work, as they have no bearing on thegeneral conclusions made here. That is why they are not pursued here. Now following Eq.(10),11 (a) r c = 0.1 J n , l ( q ) q 1s2s3s2p3p3d 0 0.66 1.32 1.98 0 0.7 1.4 2.1 (b) r c = 10 J n , l ( q ) q 1s2s3s2p3p3d (c) r c = 20 J n , l ( q ) q 1s2s3s2p3p3d FIG. 1: CP for all six n ≤ r c values 0 . , ,
20 in panels (a)-(c). See text for details. one can easily write, (cid:28) p (cid:29) s = 83 πZ = 0 . Z , (cid:28) p (cid:29) p = 6415 πZ = 1 . Z , (cid:28) p (cid:29) d = 3072525 πZ = 1 . Z , (cid:28) p (cid:29) f = 163842205 πZ = 2 . Z , (cid:28) p (cid:29) g = 13107214553 πZ = 2 . Z . (36)Now the focus is to investigate E cn,ℓ and S cn,ℓ of FHA. The ground-state E c , and S c , possessfollowing closed forms (details are provided in Appendix I & II respectively). E c , = ω = (cid:18) π (cid:19) Z B (cid:18) , (cid:19) = (cid:18) π (cid:19) Z Γ( )Γ( )Γ(11) = (cid:20) πZ (cid:21) (37) S c , = 12 ln 24 π + 12 ln Z − . (38)Finally, when Z = 1, Eqs. (37), (38) produce E c s = π = 0 . S c s = ln 24 π − =0 . S cn,ℓ , E cn,ℓ in 1 s state only. But for higher states,it is difficult to derive their closed forms. However, numerical values can be achieved. Results forsome of these states are provided in Table II of Sec. III.B. These outcomes in 1 s provide us an ideato predict the general trend in S cn,l , E cn,l with respect to Z . Later, Sec. III.C establishes that S cn,l linearly increases with Z , whereas, E cn,l is inversely proportional to Z .12 . Confined H atom Now we shift our focus to the central theme of this work, viz. ,CHA. The numerically calculatedCPs, obtained from Eq. (9), are depicted in Fig. 1. The six lowest states (1s, 2s, 3s, 2p, 3p,3d) having principal quantum number n ≤ r c values 0.1, 10, 20, identifying strong,intermediate and weak confinement regions are offered in panels (a)-(c) respectively. Accuracy ofthese CPs have been thoroughly verified by independently computing selected expectation values, D p E , h p i and h p i in accordance with the right side of Eq. (10). The first one is directly connectedto Compton profile, whereas h p i is proportional to kinetic energy. Similarly, h p i is related torelativistic kinetic energy. The results obtained using Eq. (10) are collected in Table I, for all sixstates at seven different box radii, namely, 0 . , . , . , , , , ∞ . It is a known fact that, thesemomentum moments can also be computed by either (a) using eigenfunction (Eq. (3)) in r spaceor (b) employing EMD in p space. The results obtained from Eq. (10) are in complete agreementwith the reference values achieved for both free and confined H atom by adopting the above twomethods. Due to the lack of space, only 1 s state results of CHA are reported in the footnote. Allthe quantities in this table match up to all the decimal points reported. This convergence acts asa probe about the accuracy of the constructed CPs.An in-depth analysis of panels (a)-(c) of Fig. 1 suggests that, there appears multiple humpsin a given CP, which become prominent with rise in r c . Interestingly, in a particular state, thenumber of such humps corroborate the number of radial nodes in that state. Additionally, thesharpness of these CPs enhances with weakening of pressure effect. Similarly a careful examinationof Table I reveals that, for a given state characterized by quantum numbers ( n, ℓ ), the initialintensity D p E n,ℓ = J n,ℓ ( q = 0) tends to grow with r c , while both h p i n,ℓ and h p i n,ℓ decay. Thedependence of h p i n,ℓ and h p i n,ℓ on n, ℓ quantum number at free and confined condition is very clearand supports our previous observation [42]. It may be recalled that, in FHA h p i is independentof ℓ and decays with rise in n . But in CHA it varies with both n, ℓ ; at a fixed ℓ , it grows with n ,whereas at a given n , decreases as ℓ progresses. In contrast, h p i , in both CHA and FHA, dependson n, ℓ , producing similar pattern as found here.In order to get a further insight, dependency of D p E n,ℓ on n, ℓ quantum numbers is addressedin Fig. 2. It may be noted that D p E and h p m i are inversely proportional to each other. Therefore,this particular study will automatically comment about the qualitative nature of h p m i with changein n, ℓ . From panel (a), it is seen that, at strong confinement region ( r c = 0 . ℓ initialintensity falls off at the end with rise in n . However, at a fixed n , it advances with rise in l .13 ABLE I: h p i , h p i , h p i for six low-lying states of CHA at seven r c . See text for detail. State Property r c = 0 . r c = 0 . r c = 0 . r c = 1 r c = 5 r c = 10 r c = ∞h p i § h p i † h p i ‡ h p i h p i h p i h p i h p i h p i h p i § h p i h p i h p i h p i h p i h p i § h p i h p i † Near-exact values of h p i for 1 s state in CHA obtained by using Eq. (3) at r c = 0 . , . , . , , , , ∞ are: 987 . , . , . , . , . , . , ‡ Near-exact values of h p i for 1 s state in CHA obtained by using Eq. (3) at r c = 0 . , . , . , , , , ∞ are: 975768 . , . , . , . , . , . , § Exact values in FHA, from Eq. (36), for 1s, 2p, 3d states are: 0 . , . , . Consequently, with increase in radial nodes (so does kinetic energy), D p E decreases and vice-versa . On the contrary, as r c → ∞ in panel (d), a completely opposite trend of D p E with respectto n, ℓ emerges, where increase in radial nodes leads to the accumulation of D p E . But as is known,in FHA, kinetic energy depends only on n (cid:0) Z n (cid:1) . Moreover, panels (b) and (c) show the appearanceof a maximum point, whose positions shift to right as r c goes up. Actually, these two segmentsserve as a missing link in the evolution of H atom from CHA to FHA.From the foregoing discussion of Figs. 1 and 2 as well as Table I, it is quite clear that theconstructed CPs presented above are sufficiently accurate. Several interesting features may beemphasized, viz. , (i) enhancement of sharpness of CP’s with relaxation of confinement (ii) alterationof behavioral pattern of momentum moments from CHA to FHA (iii) influence of radial nodes onCP. But the reasons behind these are not clear. Moreover, this analysis is inadequate to explainthe significance of such broadening/sharpening of CP with pressure. In this context, information14 c = 20 〈 / 〉 n , l l=0l=1l=2l=3l=4 c = ∞ 〈 / 〉 n , l l=0l=1l=2l=3l=4 c = 0.1 〈 / 〉 n , l n l=0l=1l=2l=3l=4 c = 10 〈 / 〉 n , l n l=0l=1l=2l=3l=4 FIG. 2: D p E n,ℓ versus n for s, p, d, f, g states of CHA, at four r c ’s in panels (a)-(d). See text for details. based measures like S, E may serve our purpose. Estimation of these two quantities may providethe required opportunity for CP in CHA, which we will be pursued now.Both S cn,ℓ and E cn,ℓ values for six states (same as Table I) of CHA, at ten different r c ’s,(0 . , . , . , , . , , , , , ∞ ) are given in Table II. In all cases, S cn,ℓ regresses and E cn,ℓ ad-vances with progress in r c . Since S represents the measure of uncertainty in a given distribution,a sharp distribution has less uncertainty and hence low S and a broad distribution corresponds tolarger uncertainty and hence greater S . Therefore, for an arbitrary-( n, ℓ ) state, CP gets flattenedwith rise in pressure and conversely, with increase in cavity radius, CP becomes sharp. Thus, atstrong confinement region, we get broader profile having higher uncertainty in kinetic energy aswell as in momentum. Additionally at high pressure regime, boundedness of an electron enhances15 ABLE II: S cn,l , E cn,l for 1s, 2s, 3s, 2p, 3p, 3d states in CHA at ten r c ’s. See text for detail. r c S cn,l
1s 2s 3s 2p 3p 3d0.1 2.14595 2.48251 2.67471 2.32217 2.58115 2.441890.2 1.79988 2.13588 2.32852 1.97559 2.23462 2.095310.5 1.34408 1.67826 1.87058 1.51781 1.77381 1.636271 1.00466 1.33280 1.52472 1.17187 1.43015 1.290642.5 0.59679 0.87982 1.06517 0.71811 0.97342 0.833385 0.42686 0.51831 0.73501 0.38977 0.63244 0.489998 0.41196 0.22351 0.49872 0.20212 0.40834 0.2633010 0.41188 0.09081 0.37264 0.13676 0.30206 0.1608112 0.41174 0.00438 0.25721 0.10534 0.21138 0.08282 ∞ ¶ − − − − E cn,l ∞ ¶ ¶ Exact values of S cn,ℓ , E cn,ℓ for 1s state in FHA, from Eqs. (51) and (46) are: 0.411391858 and 0.27852115 respectively. leading to increment in momentum uncertainty. Moreover, the rate of dissipation of Compton en-ergy is slower in CHA compared to FHA. Note that CPs relate to kinetic energy dissipation curves;sharp CPs (less uncertainty in momentum) release kinetic energy faster compared to a broader CP(higher uncertainty momentum). Finally, one can conjecture that, a state having higher kineticenergy (implies larger S c and a resultant broad CP) releases Compton energy slowly.Next the focus is to understand the effect of state indices n, ℓ on S c , which can help to interpretthe influence of radial nodes on CPs. This can be discerned from Fig. (3), where S c plots aredisplayed for ℓ = 0 − r c values 0.1 (a), 10 (b), 20 (c), and ∞ (d), with n up to9. This first panel imprints that, at strong confinement regime ( r c = 0 . S cn,ℓ grows with n , fora given ℓ ; however at a fixed n , it diminishes as ℓ goes up. That effectively implies, an increasein node in a system leads to growth of S c and consequently the broadening in CP. Moreover, atfixed radial node condition, states with higher n possesses higher uncertainty in momentum. For16 c = 20 S c n , l l=0l=1l=2l=3l=4 -0.8-0.4 0 0.4 (d) r c = ∞ S c n , l l=0l=1l=2l=3l=4 c = 0.1 S c n , l n l=0l=1l=2l=3l=4 c = 10 S c n , l n l=0l=1l=2l=3l=4 FIG. 3: S cn,ℓ versus n for s, p, d, f, g states of CHA, at four r c ’s in panels (a)-(d). See text for details. example, a 2p state imprints broader CP compared to a 1s state. This leads to an importantconclusion that the effect of confinement on a given state is completely governed by its principalquantum number and number of nodes. In CHA, both S c as well as kinetic energy increase with n ,leading to flattening of CP. Finally, the rate of release of Compton energy falls off as n and nodesincrease. In contrast, in panel (d), an exactly opposite trend of S c is witnessed, for the limitingcase of FHA, where it reduces with increase in number of nodes. In FHA due to the presence ofaccidental degeneracy (fixed n ) the kinetic energy is independent of ℓ . But its uncertainty decreaseswith rise in ℓ . However, at a fixed n , S c increases with ℓ . It is a contradictory fact relative to CHA,where an enhancement in S c always stipulates the rise in uncertainty in both kinetic energy andmomentum. Therefore, in FHA CP gets flattened with decrease in number of nodes. Similarly,at fixed ℓ , S c abates with progress in n , explaining the increase in sharpness in CPs with rise in17 c = 20 E c n , l l=0l=1l=2l=3l=4 c = ∞ E c n , l l=0l=1l=2l=3l=4 c = 0.1 E c n , l n l=0l=1l=2l=3l=4 c = 10 E c n , l n l=0l=1l=2l=3l=4 FIG. 4: E cn,ℓ versus n for s, p, d, f, g states of CHA, at four r c ’s in panels (a)-(d). See text for details. nodal structure. This, investigation clearly suggests that, in both FHA and CHA, CP can only becharacterized and interpreted by invoking S c . Therefore, this exploration establishes the role of S c as a descriptor in both CHA and FHA. It is is a known fact that, while moving from cha TO FHA,behavorial pattern gets reversed. It can be explained by pointing out that, in FHA an increasein number of nodes shifts an orbital more away from the nucleus. However, in CHA rise in nodalstructure squeezes the orbital more towards the centre. The remaining two intermediate panels(b), (c) suggest the appearance of minima points in these plots, which, for a given state, movestowards right as r c enhances. Note that, in this figure, the minima are observable only for 1s and2p; for other states they become visible only at some sufficiently large r c , which is not approached.This again concludes that, they act as bridge between CHA and FHA.In order to revalidate the inferences drawn in Fig. 3, we have plotted E cn,ℓ of s, p, d, f, g states18 (a) r c = 0.1 - J n , l ( q ) l n J n , l ( q ) q 1s2s3s2p3p3d (b) r c = 10 - J n , l ( q ) l n J n , l ( q ) q 1s2s3s2p3p3d -1-0.5 0 0.5 0 0.6 1.2 1.8 (c) r c = 20 - J n , l ( q ) l n J n , l ( q ) q 1s2s3s2p3p3d FIG. 5: Shannon entropy density, ( − J n,ℓ ( q ) ln J n,ℓ ( q )), for all n ≤ r c = 0 . , , , in panels (a)-(c) respectively. See text for details. (for n up to 9) at four representative r c ’s namely 0 . , , , ∞ . Four panels (a)-(d) in Fig. 4 exhibitthe same. As usual in all these four r c , E cn,ℓ displays exact opposite behavior of S cn,ℓ of previousfigure. As presumed, at r c = 0 . E cn,ℓ decreases with rise in both n and radial nodesrespectively. On the contrary, panel (d) shows that, in case of FHA it increases for the same.This again reinforce the concept that (i) in CHA, pressure effect magnifies with growth in n andnodes (ii) rate of release in Compton energy reduces with r c . In two middle panels (b) and (c),a maximum point is seen. Like S cn,ℓ , this maximum point moves to right for greater r c . It is alsoevident that, the qualitative behavior of E cn,ℓ and D p E n,ℓ resemble each other.Finally, in panels (a)-(c) of Fig. 5, Shannon entropy density, ( − J ( q ) n,l ln J ( q ) n,l ), has beenplotted at three representative r c (0.1, 10, 20) respectively. Moving from left to right panels, theintensity of a given curve (for a fixed n, ℓ state) amplifies. Therefore, an increase in r c reduces thearea under a given density curve. This was clearly manifested in first section of Table II, where S cn,l for a given state decayed with growth in r c . At small r c region in panel (a), like CP, thereappears several numbers of plateau in these curves. As usual, in a given state, number of suchhumps corresponds to the number of nodes. In moderate to large r c regions (b,c) the pattern ofdensity curves changes. In left panel, for all six states, global maximum was located at q = 0. Butpositions of such minima moves towards right direction in other two panels. Moreover, the numberof humps in a given curve does not tally with number of nodes in that state.19 ABLE III: S c and E c in the ground state of He + , Li , Be , B ions at six r c . See text for details. r c S c E c Z = 2 Z = 3 Z = 4 Z = 5 Z = 2 Z = 3 Z = 4 Z = 50.1 1.69065 1.89338 2.03723 2.14880 0.007945 0.00793 0.00791 0.007890.5 0.94337 1.14610 1.28994 1.40151 0.03881 0.03799 0.03694 0.035641 0.77344 0.97617 1.12001 1.23158 0.073875 0.06822 0.06093 0.053022 0.75799 0.96082 1.10499 1.21666 0.12186 0.09142 0.06957 0.0557210 0.75796 0.96070 1.10458 1.21641 0.139345 0.09290 0.06967 0.05574 ∞ C. Confined Hydrogen isoelectronic series
The effect of Z on confinement can easily be understood by analyzing Eqs. (20), (22) and (23).Let us assume that, both ¯ h = 1 and m = 1. Therefore, the moments becomes lead to, (cid:28) p (cid:29) n,ℓ = 1 Z (cid:28) p (cid:29) n,ℓ , h p m i n,ℓ = Z m h p m i n,ℓ , (39)whereas the information entropies, in this case, turn out to be, S cn,ℓ (1 , Z, r c , q ) = 12 ln Z + S cn,ℓ (1 , , Zr c , q ) , E cn,ℓ (1 , Z, r c , q ) = 1 Z E cn,ℓ (1 , , Zr c , q ) . (40)Equation (39) suggests that, as one passes from Z = 2 to 4, D p E n,ℓ lessens while h p m i n,ℓ goesup. It is a known fact that, a rise in h p m i n,ℓ is always accompanied by a drop in D p E n,ℓ and vice-versa . Like other instances, here also J n,ℓ ( q = 0) will provide a smooth monotonic decreasingcurve, if plotted against |E| [8]. Moreover, it is seen from Eq. (40) that, S cn,ℓ progresses and E cn,ℓ regresses with growth in Z . That means, the intensity of CP abates with rise in Z , which enhancesthe boundedness of electron. The ground-state S c , E c of He + , Li , Be , B are reported inTable III at six distinct r c values, i.e., 0.1, 0.5, 1, 2, 10 and ∞ . These results demonstrate thevalidity of expressions given in Eqs. (40); in all four occasions, S c advances and E c declines withlengthening of box radius. Therefore, with relaxation in confinement, the rate of dissipation inkinetic energy magnifies. However, an enhancement in Z value, inhibits the process. IV. FUTURE AND OUTLOOK
Within the impulse approximation, CPs for confined H-like atoms, has been presented, forthe first time. The accuracy and reliability has been verified by calculating several momentummoments. Besides, S c and E c were also invoked in a novel way to analyze the CPs–their correctness20uggests the impulse approximation to hold good in confined conditions. Such calculations are alsoperformed in the respective unconfined systems. To the best of our knowledge, this is the firstundertaking of such calculations. Moreover, they are found to play the role of good descriptors , asthey offer a proper interpretation about the boundedness of electron, and influence of radial nodeon a given state. As an offshoot, several interesting analytical relations involving S c , E c , D p E , h p m i ,with Z , have been derived. It is observed that, the effect of Z on CP’s remains similar in bothconfined and free conditions.Experimental verification of these results is highly desirable. This will open up a new di-mension in the high-pressure physics and chemistry. It is well known that, confined systems aregenerally, not exactly solvable; and moreover, accurate calculation of energy and density may bequite demanding and challenging for conventional theoretical approximations. However, once CP isavailable experimentally, one can follow the usual route used in free condition to construct EMD,and subsequently wave function and energy. This may be an interesting recipe to explain thebonding pattern, coordination number and reactivity in such stressed systems. Also, a theoreticalexploration in many-electron systems would be quite helpful. Similar study for confined molecularsystems may provide vital insight about the effect of confinement in chemical bonding. V. ACKNOWLEDGEMENT
Financial support from BRNS, India (sanction order: 58/14/03/2019-BRNS/10255) is gratefullyacknowledged. NM thanks CSIR, New Delhi, India, for a Senior Research Associateship (Pool No.9033A). Critical constructive comments from an anonymous referee is greatly appreciated.
VI. APPENDIX I: ONICESCU ENERGY CALCULATION
At first let us work out the α -order entropic moment, ω α . It will provide an opportunity tocalculate an arbitrary-order entropic moments. It can be derived as follows, ω α = Z ∞ (cid:18) πZ (cid:19) α h(cid:0) qZ (cid:1) + 1 i α d q. (41)Assuming qZ = u , this equation can be transformed in to, ω α = Z ∞ (cid:18) π (cid:19) α Z − α u + 1] α d u. (42)21urther substitution of u = tan θ in Eq. (42) produces, ω α = Z π (cid:18) π (cid:19) α Z − α sec θ [tan θ + 1] α d θ (43)= (cid:18) π (cid:19) α Z − α Z π cos α − θ d θ. (44)Now, making use of the standard integral, R π cos ( µ − θ d θ = 2 ( µ − B (cid:0) µ , µ (cid:1) ( µ = 6 α − ω α = (cid:18) π (cid:19) α Z − α (6 α − B (cid:18) (6 α − , (6 α − (cid:19) . (45)It is needless to mention that, at α = 2, this entropic moment in Eq. (45) reduces to E cn,ℓ . Therefore, E c , = ω = (cid:18) π (cid:19) Z B (cid:18) , (cid:19) = (cid:18) π (cid:19) Z Γ( )Γ( )Γ(11) = (cid:20) πZ (cid:21) (46) VII. APPENDIX II: SHANNON ENTROPY CALCULATION
In a similar fashion, S cn,l for 1 s state will take the form, S c , = − Z ∞ (cid:18) πZ (cid:19) h(cid:0) qZ (cid:1) + 1 i ln (cid:18) πZ (cid:19) h(cid:0) qZ (cid:1) + 1 i d q. (47)Once again putting qZ = u and u = tan θ , Eq. (47) can be recast in to, S c , = (cid:20) ln (cid:18) π (cid:19) + ln Z (cid:21) (cid:18) π (cid:19) Z π cos θ d θ − π Z π cos θ ln cos θ d θ,S c , = (cid:20) ln (cid:18) π (cid:19) + ln Z (cid:21) (cid:18) π (cid:19) A − π A . (48)Now invoking the following standard integrals [71], Z π cos m θ d θ = (2 m − m )!! π , Z π cos k θ ln cos θ d θ = − (2 k − k k ! π " ln 2 + k X m =1 ( − m m , (49)we obtain A , A as, A = Z π cos θ d θ = 3 π ,A = Z π cos θ ln cos θ d θ = − π (cid:20) ln 2 − (cid:21) . (50)22inally inserting the values of A and A in Eq. (48) one gets, S c , = 12 ln 24 π + 12 ln Z − . (51) [1] A. H. Compton, Phys. Rev. , 409 (1923).[2] J. Lahtola, M. Hakala, J. Vaara and K. H¨am¨al¨ainen, Phys. Chem. Chem. Phys. , 5630 (2011).[3] P. M. Platzman and N. Tzoar, Phys. Rev. , 410 (1965).[4] P. Eisenberger and P. M. Platzman, Phys. Rev. A , 415 (1970).[5] I. R. Epstein, Phys. Rev. A , 160 (1973).[6] M. J. Cooper, P. E. Mijnarends, N. Shiotani, N. Sakai and A. Bansil, X-ray Compton Scattering , volume5, Oxford Series on Synchrotron Radiation, (Oxford University Press, 2004).[7] M. J. Cooper, Rep. Prog. Phys. , 415 (1985).[8] S. R. Gadre and S. B. Sears, J. Chem. Phys. , 4321 (1979).[9] S. B. Sears and S. R. Gadre, J. Chem. Phys. , 4626 (1981).[10] S.-K. Son, O. Geffert and R. Santra, J. Phys. B , 064003 (2017).[11] J. T. Okada, P. H.-L. Sit, Y. Watanabe, B. Barbiellini, T. Ishikawa, Y. J. Wang, M. Itou, Y. Saku-rai, A. Bansil, R. Ishikawa, M. Hamaishi, P.-F. Paradis, K. Kimura, T. Ishikawa, and S. Nanao,Phys. Rev. Lett. , 698 (2011).[13] W. B. Duncanson and C. A. Coulson, Proc. Phys. Soc. , 190 (1945).[14] M. Hakala, S. Huotari, K. H¨am¨al¨ainen, S. Manninen, Ph. Wernet, A. Nilsson and L. G. M. Pettersson,Phys. Rev. B , 125413 (2004).[15] J. R. Hart and A. J. Thakkar, Int. J. Quantum Chem. , 673 (2005).[16] A. J. Thakkar, J. W. Liu and W. J. Stevens, Phys. Rev. A , 4695 (1986).[17] A. J. Thakkar, J. W. Liu and G. C. Lie, Phys. Rev. A , 5111 (1987).[18] A. J. Thakkar and H. Tatewaki, Phys. Rev. A , 1336 (1990).[19] M. M´erawa, M. R´erat and A. Lichanot, Int. J. Quantum Chem. , 63 (1999).[20] S. Ragot, J. Chem. Phys. , 014106 (2006).[21] Y. Kubo, J. Phys. Chem. Solids , 2202 (2005).[22] C. Pisani, M. Itou, Y. Sakurai, R. Yamaki, M. Ito, A. Erba and L. Maschio,Phys. Chem. Chem. Phys. , 933 (2011).[23] M. J. Cooper, Adv. Phys. , 453 (1971).[24] M. J. Cooper, Radiat. Phys. Chem. , 63 (1997).
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