Accurate computations of bound state properties in three- and four-electron atomic systems in the basis of multi-dimensional gaussoids
aa r X i v : . [ phy s i c s . a t m - c l u s ] J u l Accurate computations of bound state properties in three- andfour-electron atomic systems in the basis of multi-dimensionalgaussoids.
Alexei M. Frolov ∗ Department of Applied Mathematics, University of Western Ontario,London, Ontario, N6H 5E5, Canada
David M. Wardlaw † Department of Chemistry, Memorial University of Newfoundland,St.John’s, Newfoundland and Labrador, A1C 5S7, Canada (Dated: August 30, 2018)
Abstract
Results of accurate computations of bound states in three- and four-electron atomic systems arediscussed. Bound state properties of the four-electron lithium ion Li − in its ground 2 S − stateare determined from the results of accurate, variational computations. We also consider a closelyrelated problem of accurate numerical evaluation of the half-life of the beryllium-7 isotope. Thisproblem is of paramount importance for modern radiochemistry. ∗ E–mail address: [email protected] † E–mail address: [email protected] . INTRODUCTION In this communication we consider the bound states properties of the negatively chargedLi − ion in its ground 2 S ( L = 0) − state, or 2 S − state, for short. It is well known that the2 S − state is the only bound state in this ion. The electronic structure of this state in Li − corresponds to the 1 s s electron configuration. The negatively charged lithium ion hasbecome of interest in numerous applications, since formation of these ions is an importantstep for workability of lithium and/or lithium-ion electric batteries (see, e.g., [15], [16] andreferences therein). Both lithium and lithium-ion batteries are very compact, relativelycheap and reliable sources of constant electric current which are widely used in our everydaylife and in many branches of modern industry. However, it appears that the Li − ion is not awell studied atomic system. Indeed, many bound state properties of this ion have not beenevaluated at all even for an isolated Li − ion in vacuum. In reality, it is crucial to know itsbound state properties in the presence of different organic acids which are extensively usedin lithium-ion batteries.Our goal in this study is to determine various bound state properties of the four-electron(or five-body) Li − ion and compare them with the corresponding properties of the neutralLi atom in its ground 2 S − state. It should be mentioned that many of the bound stateproperties of the Li − ion have not been evaluated in earlier studies. The negatively chargedLi − ion is here described by the non-relativistic Schr¨odinger equation H Ψ = E Ψ, where H is the non-relativistic Hamiltonian, E ( <
0) is the eigenvalue and Ψ is the bound state wavefunction of the Li − ion. Without loss of generality we shall assume that the bound statewave function Ψ has the unit norm. The non-relativistic Hamiltonian H of an arbitraryfour-electron atom/ion takes the form (see, e.g., [3]) H = − ¯ h m e h ∇ + ∇ + ∇ + ∇ + m e M ∇ i − Qe X i =1 r + e X i =1 4 X j =2( j>i ) r ij , (1)where ¯ h = h π is the reduced Planck constant, m e is the electron mass and e is the electriccharge of an electron. In this equation and everywhere below in this study the subscripts 1,2, 3, 4 designate the four atomic electrons e − , while the subscript 5 (= N ) denotes the heavyatomic nucleus with the mass M ( M ≫ m e ), and the positive electric (nuclear) charge is Qe .The notation r ij = | r i − r j | = r ji in Eq.(1) and everywhere below stands for the interparticledistances between particles i and j . These distances are also called the relative coordinates2o differentiate them from the three-dimensional coordinates r i , which are the Cartesiancoordinates of the particle i . In Eq.(1) and everywhere below in this work we shall assumethat ( ij ) = ( ji ) = (12), (13), (14), (15), (23), (24), (25), (34), (35) and (45), for four-electronatomic systems and particle 5 means the atomic nucleus. Analogously, for three-electronatomic systems we have ( ij ) = ( ji ) = (12), (13), (14), (23), (24) and (34), where particle4 is the atomic nucleus. Below only atomic units ¯ h = 1 , | e | = 1 , m e = 1 are employed. Inthese units the explicit form of the Hamiltonian H , Eq.(1), is simplified and takes the form H = − h ∇ + ∇ + ∇ + ∇ + m e M ∇ i − Q X i =1 r i + X i =1 4 X j =2( j>i ) r ij , (2)where Q is the nuclear charge of the central positively charged nucleus. For the negativelycharged Li − ion we have Q = 3. Note that the stability of the bound 2 S − state in the Li − ion means stability against its dissociation (or ionization) Li − → Li(2 S ) + e − , where thenotation Li(2 S ) means the lithium atom in its ground (doublet) 2 S − state. In general,the bound state properties of the neutral Li atom in its ground 2 S − state are importantto predict and approximately evaluate analogous bound state properties of the negativelycharged Li − ion. In this study such evaluations are considered carefuly, but first of allwe need to describe our method which is used to construct accurate wave functions forfour-electron atomic systems. This problem is considered in the next Section. II. VARIATIONAL WAVE FUNCTIONS FOR FOUR- AND THREE-ELECTRONATOMIC SYSTEMS
To determine accurate solutions of the non-relativistic Schr¨odinger equation H Ψ = E Ψin this study we apply variational expansion written in multi-dimensional gaussoids. Eachof these gaussoids explicitly depends upon a number of the relative coordinates r ij . For four-electron atoms and ions there are ten relative coordinates: r ij = r , r , r , r , r , . . . , r .In particular, for the singlet S ( L = 0) − states in four-electron atomic systems the variationalexpansion in multi-dimensional gaussoids takes the form (see, e.g., [4], [5]): ψ ( L = 0; S = 0) = N A X i =1 C i A [exp( − α ij r ij ) χ (1) S =0 ] + N B X i =1 G i A [exp( − β ij r ij ) χ (2) S =0 ] (3)where A is the complete four-electron anti-symmetrizer, C i (and G i ) are the linear vari-ational coefficients of the variational function, while α ij , where (ij) = (12), (13), . . . , (45),3re the ten non-linear parameters in the radial function associated with the χ (1) S =0 spin func-tion. Analogously, the notation β ij stands for other ten non-linear parameters in the radialfunction associated with the χ (2) S =0 spin function. Note that these two sets of non-linearparameters α ij and β ij must be varied independently in calculations. Notations χ (1) S =0 and χ (2) S =0 in Eq.(3) designate the two independent spin functions which can be considered forthe singlet 2 S − state, or (2 S | s s ) − electron configuration. The explicit forms of thesetwo spin functions are: χ (1) S =0 = αβαβ + βαβα − βααβ − αββα (4) χ (2) S =0 = 2 ααββ + 2 ββαα − βααβ − αββα − βαβα − αβαβ (5)where α and β are the single-electron spin-up and spin-down functions [6]. In numericalcalculations of the total energies and other spin-independent properties (i.e. expectationvalues) one can always use just one spin function, e.g., χ (1) S =0 from Eq.(4). It follows from thefact that the Hamiltonian Eq.(2) does not depend explicitly upon the electron spin and/orany of its components.For three-electron atomic systems considered in this study, e.g., for the Li-atom, theanalogous expansion in multi-dimensional gaussoids is written in the form [4], [5] ψ ( L = 0; S = 12 ) = N A X i =1 C i A [exp( − α ij r ij ) χ (1) S = ] + N B X i =1 G i A [exp( − β ij r ij ) χ (2) S = ] (6)where A is the complete three-electron (or three-particle) anti-symmetrizer, C i (and G i )are the linear variational coefficients of the variational function, while α ij , where (ij) =(12), (13), . . . , (34), are the six non-linear parameters for three-electron atomic systems.In these notations the notations/indexes 1, 2, 3 designate three atomic electrons, while 4means heavy atomic nucleus. Analogously, the notation β ij stands for other six non-linearparameters which must also be varied (independently of α ij ) in calculations. Notations χ (1) S = and χ (2) S = in Eq.(3) designate the two independent spin functions which can be consideredfor the doublet 2 S − state, or (2 S | s s ) − electron configuration. The explicit forms ofthese two spin functions are: χ (1) S = = αβα − βαα (7) χ (2) S = = 2 ααβ − βαα − αβα (8)4he Hamiltonian of the three-electron atomic system (e.g., Li-atom) is H = − h ∇ + ∇ + ∇ + m e M ∇ i − Q X i =1 r i + X i =1 3 X j =2( j>i ) r ij , (9)where all notations have the same meaning as in Eq.(2). The only difference with Eq.(2) isthe fact that here we are dealing with the three-electron atomic systems. In particular, theindex 4 means the heavy atomic nucleus with the electric charge Q (or Qe ). Note also thatthe explicit forms of the three- and four-particle anti-symmetrizers, optimization of the non-linear parameters and other important steps in construction of the variational expansionsEqs.(3) - (6) for four- and three-electron atoms, respectively, have been described in detailin a large number of papers (see, e.g., [7] - [9], [10] and references therein). Here we do notwant to repeat these descriptions of the four- and three-electron variational methods whichare used for accurate numerical calculations of various few-electron atoms and ions. In thenext two Sections we discuss results obtained for the negatively charged four-electron Li − ion and neutral three-electron Li atom, respectively. III. RESULTS FOR THE NEGATIVELY CHARGED LITHIUM ION
As mentioned above in this paper we consider the ground 2 S − state of the Li − ion withthe infinitely heavy nucleus (i.e., the ∞ Li − ion). Our goal is to determine the total energyof this ion and expectation values of some of its properties. Such properties include a fewpowers of interparticle distances h r nij i , where n = − , − , , , , n = 0 each of these ex-pectation values equals unity), electron-nucleus and electron-electron delta-functions, singleelectron kinetic energy h p e i , and a few others. As shown in the Appendix the electron-nucleus and electron-electron kinetic correlations h p e · p N i and h p e · p e i are not truly inde-pendent atomic properties. Therefore, there is no need to include those expectation valuesin Table I. Table I also includes the bound state properties of the ground 2 S − state in theneutral Li atom, which is a three-electron atomic system. All these properties are expressedin atomic units.The expectation values of the different bound state properties computed for the four-electron Li − ion (or ∞ Li − ion) can be compared with the similar properties of the ground2 S − state of the three-electron Li atom (or ∞ Li atom). As follows from Table I there aresome substantial differences in the electron-nucleus and electron-electron distances h r eN i and5 r ee i in the four-electron Li − ion and three-electron Li atom. For the Li − ion these distancesare significantly larger than for the neutral Li atom. The same conclusion is correct forall positive powers of these inter-particle distances, i.e. for the h r keN i and h r kee i expectationvalues (here k is integer and k ≥ h r keN i and h r kee i expectation values (here k is integer and k ≤ −
1) the situation isopposite. This is an indication of the known fact that the Li − ion is a weakly-bound, four-electron system atomic system. This fact can be confirmed by calculation of the followingdimensionless ratio ǫ = E (Li − ) − E (Li) E (Li − ) ≈ . E (Li − ) and E (Li) are the total energies of the negatively charged Li − ion in the ground2 S − state and Li atom in the ground 2 S − state. A very small value of this parameter ǫ ,which here is significantly less that 0.01 (or 1 %), is a strong indication that the Li − ion is anextremely weakly-bound atomic system. This allows one to represent the internal structureof the bound state in the Li − ion as a motion of one electron in the ‘central’ field created bythe infinitely heavy Li atom. In other words, the electronic structure of this ion is 1 s s andone of the two outer-most electrons moves at very large distances from the central nucleus.In reality, this representation is only approximate, since, e.g., there is an exchange symmetrybetween two electrons in the 2 s shell. Nevertheless, such a ‘cluster’ structure can be usefulto predict and explain a large number of bound state properties of the Li − ion. For instance,consider the expectation value of the inverse electron-nucleus distance, i.e. h r − eN i . From thedefinition of this expectation value we write the following expression h r − eN i = 14 (cid:16) h r − N i + h r − N i + h r − N i + h r − N i (cid:17) (11)where all expectation values in the right-hand side are determined without any additionalsymmetrization between four electrons. As mentioned above the Li − ion has a sharp clusterstructure and its fourth electron is located on avarage far away from the central nucleus.This means that h r − N i ≈
0. In this case it follows from Eq.(11) that h r − eN i = 34 h r − N i = 34 h r − eN i ≈ h r − eN i Li (12)where h r − eN i Li is the corresponding expectation value for the neutral Li-atom. It is clearthat this equality is only approximate. Analogous approximate evaluations can be obtained6or some other properties, e.g., for the expectation values of all delta-functions and inversepowers of electron-nucleus and electron-electron distances.Table I contains a large number of bound state properties of the negatively charged Li − ion. Numerical values of these properties are of interest in various scientific and techni-cal applications, including quite a few applications to electro-chemistry of the lithium andlithium-ion batteries. Our expectation values form a complete set of numerical values whichcan be useful in analysis of different macroscopic systems containing neutral lithium atomsand negatively charged lithium ions. IV. ACCURATE COMPUTATIONS OF THE GROUND STATES IN HEAVYTHREE-ELECTRON IONS
For three-electron atoms and ions one finds a large number of interesting problems whichhave not been solved in earlier studies. Here we consider the two following problems: (1) ac-curate computations of the ground state (2 S − state) energies for some heavy three-electronions (Sc - Ni ), and (2) accurate numerical evaluation of some basic geometrical prop-erties (expectation values) for these three-electron ions. In these computations we haveassumed that all atomic nuclei are infinitely heavy. Results of our computations of theseions (ground doublet 2 S − states) can be found in Table II (in atomic units). It should bementioned that the overall accuracy of the variational expansion of six-dimensional gaus-soids is substantially greater than the analogous accuracy achieved with a similar variationalexpansion for the four-electron atoms/ions. In reality, the accuracy of our procedure hasbeen restricted by the double-precision accuracy of our optimization code and results in amaximal accuracy for the total energy of 3 · − − · − a.u . This maximal accuracy wasobserved in bound state calculations of the heavy ions (all three-electron ions after Cl ).Our current results (energies) obtained for heavy three-electron ions allow us to completethe Table (published in [11]) of the bound states energies of different few-electron atomicsystems (see Table III). The original Table in [11] was based on our highly accurate resultsfor two-electron systems and also on the results from [8] and [12] for three- and four-electronatoms/ions, respectively. In general, the main idea from [11] works well for few-electronatoms and ions. However, the overall accuracy of our predictions for total energies of few-electron atomic systems is not very high, since the total energies of the four-electron atoms7nd ions have been determined [12] to the accuracy which is substantially lower than theanalogous accuracy achieved for two- and three-electron atomic systems. For instance, byusing data from the last column of Table III and asympotic formulas for Q − expansion(see, e.g., [11]) one can obatin only very approximate value of the total energy of the ∞ Li − ion. Furthermore, the total energies of some four-electron ions, e.g., in the case of Ar, differsubstantially from numerical values known from other papers (see, e.g., [13]).Another aim of this study was to perform accurate numerical evaluations of bound stateproperties for a number of heavy three-electron ions. Here we chose the same multi-chargedthree-electron ions Sc - Ni in their ground doublet 2 S − states. Results of these cal-culations can be found in Table II (in atomic units), where a number of electron-electronand electron-nucleus h r kij i expectation values (for k = -2, -1 and 1) are shown. As followsfrom Table II the computed expectation values smoothly vary with the electric charge ofthe atomic nucleus Q . In other words, these expectation values are uniform functions of Q .Formally, we can propose a number of relatively simple interpolation formulas (upon Q ) forthese expectation values. V. ON THE HALF-LIFE OF THE BERYLLIUM-7 ISOTOPE
Results of our accurate computations of the ground 2 S − state in the weakly-bound Li − ionindicate clearly that our variational expansion Eq.(3) is very effective in applications to four-electron atomic systems. In this Section we apply the same variational expansion, Eq.(3),to investigate another long-standing problem known in the atomic physics of four-electronatomic systems. Briefly, our goal is to explain variations of the half-life of the beryllium-7isotope in different chemical enviroments. As follows from the results of numerous experi-ments, the half-life of the Be isotope is ‘chemically dependent’, i.e. it varies by ≈ Be) isotope. It should be mentioned that in modernlaboratories different chemical compounds containing Be atoms are not ‘exotic’ substances,since the nuclei of Be are formed in the ( p ; n ) − and ( p ; α ) − reactions of the Li and Bnuclei with the accelerated protons. A few other nuclear reactions involving nuclei of some8ight and intermediate elements, e.g., C, Al, Cu, Au, etc, also lead to the formation of Benuclei. In general, an isolated Be nucleus decays by using a few different channels, themost important of which is the electron capture (or e − − capture) of one atomic electronfrom the internal 1 s − shell. The process is described by a simple atomic-nuclear equation Be → Li, where there is no free electron emitted after the process. During this processthe maternal Be nucleus is transformed into the Li nucleus which can be found either inthe ground state, or in the first excited state. The subsequent transition of the excited Li ∗ nucleus into its ground state Li proceeds with the emission of a γ − quantum which hasenergy E γ ≈ . M eV . Such γ − quanta can easily be registered in modern experimentsand this explains numerous applications of chemical compounds of Be in radio-chemistry.Let us discuss the process of the electron capture in the Be-atom in detail. Assume for amoment that all Be atoms decay by electron capture from the ground (atomic) 2 S − state.In this case, by using the expectation value of the electron-nucleus delta-function h δ ( r eN ) i computed for the ground 2 S − state of an isolated Be-atom we can write the followingexpression for the half-life τ of the Be atom/isotope τ = 1Γ = 1 A h δ ( r eN ) i (13)where Γ is the corresponding width and A is an additional factor which in principle dependson the given chemical compound of beryllium. The half-life τ determines the moment when50 % of the incident Be will have decayed by electron capture. An analytical formula for τ ,Eq.(13), follows from the fact that the corresponding width Γ = τ − must be proportional tothe product of theexpectation value of the electron-nucleus delta-function and an additionalfactor A . The expectation value of the electron-nucleus delta-function computed with thenon-relativistic wave function determines the electron density at the surface of a spherewith the spatial radius R ≈ Λ e = ¯ hm e c a = αa , where a is the Bohr radius a ≈ ¯ h m e e ( ≈ . · − cm ), c is the speed of light and Λ e is the Compton wave length. The ‘constant’ A in Eq.(13) represents an ‘additional’ probability for an electron (point particle) to penetratefrom the distance R ≈ Λ e = αa to the surface of the nucleus R N ≈ · − cm .A numerical value of A can be evaluated by assuming that the mean half-life of the Be-atom in its ground 2 S − state equals 53.60 days and by using our best expectation valueobtained for the expectation value of the electron-nucleus delta-function h δ ( r eN ) i ≈ a.u. , one finds that Γ ≈ . · − sec − . From here we find that the factor A in9q.(13) equals A ≈ . · − h δ ( r eN ) i ≈ . · − (14)where the expectation value h δ ( r eN ) i must be taken in atomic units. As follows from numer-ous experiments the mean life-timed of chemical compounds which contain some Be-atom(s)are ≈
53 - 54 days. This means that the ‘constant’ A varies slowly in actual molecules. Thisallows us to write the following approximate formula for the ratio of half-life of the twodifferent molecules X(Be) and Y(Be) which contain Be atoms τ (X(Be)) τ (Y(Be)) = h δ ( r eN ); Y(Be) ih δ ( r eN ); X(Be) i (15)Let us apply this formula to the case when one of the Be-atoms is in the ground 2 S − state,while another such an atom is in the triplet 2 S − state. The expectation value of the δ eN -function for the ground state in the Be-atom is given above, while for the triplet state wehave h δ ( r eN ) i ≈ a.u. Both these expectation values were determined in our highlyaccurate computations of the ground 2 S − and 2 S − state in the four-electron Be atom.With these numerical values one finds from Eq.(15) that the half-life of the Be atom in itstriplet 2 S − state is 1.009794 times (or by ≈ Be atom in its ground singlet 2 S -state. This simple example includes two differentbound states in an isolated Be-atom. In general, by using the formula Eq.(15) we canapproximately evaluate the half-life of the Be atoms in different molecules and compounds.The formula Eq.(15) can be applied, e.g., to BeO, BeC , BeH and many other berylliumcompounds, including beryllium-hydrogen polymers, e.g., Be n H n for n ≈ − s snℓ (or 1 s s nℓ ), where ℓ ≥ n ≥
3. In general,such an excited state arises after excitation of a single electron from the 1 s s electronconfiguration, which correspond to the ground state, or ‘core’, for short. It is clear that thefinal 1 s s nℓ configuration is the result of a single electron excitation 2 s → nℓ . All otherstates with excitation(s) of two and more electrons from the core are unbound. In general, avery substantial contribution ( ≥
95 %) to the expectation value of the electron-nucleus delta-function comes from the two internal electrons (or 1 s − electrons) of the Be-atom. Brieflythis means that the expectation value of the electron-nucleus delta-function is almost thesame for all molecules which contain the bound Be-atom. Variations in 3 % - 6 % are possible10nd they are related with the contribution of the two outer-most electrons in the expectationvalue of the electron-nucleus delta-function h δ ( r eN ) i . As follows from computational resultsthe overall contribution from two outer-most electrons is only 3 % - 6 % of the total numericalvalue. This means that variations in the chemical enviroment of one Be atom can changethe half-life of this atom by a factor of 1.03 to 1.06 (maximum). In reality, such changes aresignificantly smaller, but they can be noticed in modern experiments.It is interesting to note that analogous result (3 % - 6 % differences as maximum) canbe predicted for other nuclear processes which are influenced by variations in the chemicalenviroment, e.g., for the excitation of the
U nucleus which also depends upon chemicalenviroment [22] - [25]. It is well known (see, e.g., [24]) that the
U nucleus has an excitedstate with the energy ≈
75 - 77 eV . There is no such level in the U, U and
Unuclei. Nuclear properties of the ground and first excited states in the
U nucleus differsubstantially. Moreover, by changing the actual chemical enviroment of the
U atom wecan change the probabilities of excitation of the central nucleus, e.g., by using differentalloys of uranium, in order to change and even control nuclear properties. For instance, thisapproach can be used to achieve and even exceed critical conditions with respect to neutronfission. Theoretical evaluations and preliminary experiments show that possible changes innuclear properties of different compounds of uranium-235 do not exceed 3 - 6 %. It is verylikely that 3 - 6 % is the upper limit of influence of atomic (and molecular) properties onthe nuclear properties of different isotopes. On the other hand, possible changes in atomicand molecular properties produced by processes, reactions and decays in atomic nuclei arealways significant.Thus, if we know the expectation value of the electron-nucleus delta-function for theberyllium-7 atom within some molecule with other chemical elements, then we can evaluatethe corresponding half-life of such an atom with respect to electron capture. Currently, how-ever, this problem can be solved only approximately, since there are quite a few difficultiesin accurate computations of complex molecules as well as in actual experiments, since, e.g.,the exact value of the constant A in Eq.(13) is not known. In other words, we cannot besure that the experimental half-life mentioned above (53.60 days) corresponds to the elec-tron capture in the ground 2 S − state of an isolated Be atom. In fact, it is not clear whatchemical compounds were used (and at what conditions) to obtain this half-life. Very likely,we are dealing with some ‘averaged’ value determined for a mixture of different molecules. It11s clear that improving the overall experimental accuracy and purity of future experimentsis critical. The accuracy of future theoretical computations could also be improved. First ofall, we need to focus on accurate expectation values of the electron-nucleus delta-function h δ ( r eN ) i , rather than just accurate values of the total energy. Then the formula, Eq.(15), canbe used to determine the actual life-times of the Be atoms, which are included in differentchemical compounds.
VI. CONCLUSION
We have considered the bound state properties of the negatively charged Li − ion in theground 2 S − state. These bound state properties are compared with the analogous propertiesof the neutral Li atom. Our analysis of the bound state properties of the Li − ion is of interestsince the formation of the negatively charged Li − ions plays an important role in modernlithium and lithium-ion batteries. Expectation values of different properties determinedin this study are sufficient for all current and anticipated future experimental needs. Asfollows from the results of our calculations the Li − ion is a weakly-bound atomic systemwhich has only one bound 2 S − state. The internal structure of this state is representedas a motion of one ‘almost free’ electron in the field of a heavy atomic cluster which is theneutral Li atom in its ground 2 S − state. The computed expectation values of the boundstate properties of the Li − ion in the ground 2 S − state and the neutral Li atom in theground 2 S − state support such a picture. Moreover, the whole internal structure of the Li − ion could be reconstructed to very good accuracy if we knew the model potential betweenan electron and neutral Li atom. This corresponds to the two-body approximation which isoften used for weakly bound few-body systems. An accurate reconstruction of such a model e − -Li interaction potential should be a goal of future research. The same model potentialcould then be used to obtain the cross-section of the elastic scattering (at relatively smallenergies) for the electron-lithium scattering.It should be mentioned that the negatively charged Li − ion is of interest for possiblecreation and observation of an unstable (three-electron) He − ion which is formed in one ofthe channels of the reaction of the Li − ion with slow neutrons, e.g., Li − + n = He − + H + + e − + 4 . M eV , (16)12reliminary evaluations indicate that the probability of formation of the He − ion in thisreaction is ≈ Li − ion with slowneutrons has a very large cross-section and it can be used to produce the negatively chargedHe − ion which is unstable and decays into the neutral He atom with the emission of oneelectron. Other approaches to create relatively large numbers of the negatively charged He − ions have failed.We also investigated the situation of experimental variations of the half-life of theberyllium-7 isotope placed in different chemical enviroments. Since the middle of the 1930’sthis interesting problem has attracted significant experimental and theoretical attention. Itis shown here that the half-life of the beryllium-7 isotope in different chemical enviromentsmay vary by 3 % - 6 % (maximum). A central computational part of this problem is to deter-mine to high accuracy the electron-nucleus delta-function of the Be-atom placed in differentmolecules, ‘quasi-metalic’ alloys and other chemical compounds. The currently achieved ac-curacy is not sufficient to make accurate predictions of the half-life of the beryllium-7 atomin many molecules. Another part of the solution is to improve the accuracy and the purity ofthe chemical enviroment in all experiments performed with different molecules which includeatoms of beryllium-7. Appendix
The expectation values h p e · p N i and h p e · p e i are not presented in Table I, since theyare not truly independent from the h p e i and h p N i expectation values which are given inTable I. Indeed, for an arbitrary K − electron atom/ion the expectation values of the scalarproducts of the vectors of electron’s momenta p i ( i = 1 , . . . , K ) with the electron’s momenta p i ( j = i and j = 1 , . . . , K ) and with the momentum of the nucleus p N are simply relatedwith the expectation values of the single-electron kinetic energy and kinetic energy of theatomic nucleus: h p i · p j i = h p · p i = 2 K ( K − h h p N i − h p e i i (17) h p i · p N i = h p · p N i = − K h p N i , (18)where K ( ≥
2) is the total number of electrons in atom, h p i · p j i is the scalar product ofthe two electron momenta ( i = j ), while h p i · p N i is the scalar product of the momenta13f the atomic nucleus and electron (with index i ). Since the electron’s indexes can bechosen arbitrarily, we can replace the scalar products in Eqs.(17) - (18) by the h p · p i and h p · p N i expectation values, respectively. In general, these two expectation values determinethe electron-electron and electron-nucleus kinematic correlations in few- and many-electronatoms. In Eqs.(17) and (18) the notations h p e i and h p N i designate the single-electronkinetic energy and kinetic energy of the atomic nucleus, respectively. Therefore, there is noneed to include the h p · p i and h p · p N i expectation values in Table I. Also, it is interestingto note that the nuclear charge Q is not included in Eqs.(17) - (18). This means that Eqs.(17)- (18) can be applied to an arbitrary K − electron atom, or positevely/negatively chargedion. For two-electron atomic systems we have K = 2 and Eqs.(17) - (18) mentioned abovetake the well known form (see, e.g., [26], [27]) h p · p i = h p N i − h p e i , h p e · p N i = h p · p N i = −h p N i (19) [1] X. Zhao, Hayner, C. M. Kung, and M. C. Kung, H. H., Advanced Energy Materials , 1079(2011).[2] J. Summerfield, Journal of Chemical Education , 453 (2013).[3] L.D. Landau and E.M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory , (3rd. ed., Perg-amon Press, Oxford (UK) (1977)).[4] N.N. Kolesnikov and V.I. Tarasov, Yad. Fiz. , 609 (1982), [Sov. J. Nucl. Phys. , 354(1982)].[5] A.M. Frolov and D.M. Wardlaw, Phys. Rev. A Phys. Rev. A , 042506 (2008).[6] P.A.M. Dirac, The Principles of Quantum Mechanics (4th ed., Oxford at the Clarendon Press,Oxford (UK) (1958)).[7] S. Larsson, Phys. Rev. , 49 (1968).[8] Z.-C. Yan, M. Tambasco and G.W.F. Drake, Phys. Rev. A , 1652 (1998).[9] A.M. Frolov, M.B.Ruiz and D.M. Wardlaw, Chem. Phys. Lett., , 191 - 200 (2014).[10] M.B.Ruiz, F. Latorre and A.M. Frolov, Advances in Quant. Chem. , 119 - 137 (2016) (seealso: J.T. Margraf, M.B. Ruiz and A.M. Frolov, Phys. Rev. A , 012505 (2013)).[11] A.M. Frolov, J. Math. Chem. , 2172 (2015).
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187 (2002).[19] M. Jaeger, S. Wilmes, V. K¨olle, and G. Staudt Phys. Rev. C , 423 (1996).[20] E.V. Tkalya, A.V. Avdeenkov, A.V. Bibikov, I.V. Bodrenko and A.V. Nikolaev, Phys. Rev. C , 014608 (2012).[21] T. Ohtsuki, K. Ohno, T. Morisato, T. Mitsugashira, K. Hirose, H. Yuki and J. Kasagi, Phys.Rev. Lett. , 252501 (2007).[22] F. Asaro and I. Perlman, Phys. Rev. , 318 (1957).[23] T. Almeida, T. Von Egidy, P.H.M. Van Assche et al, Nucl. Phys. A , 71 (1979).[24] V.I. Zhudov, A.G. Zelenkov, A.G., V.M. Kulakov, V.I. Mostovoi and B.V. Odinzov, PismaZh. Eksp. Teor. Fiz. , 549 [JETP Lett. , 516 (1979)].[25] A.M. Frolov, Radiation Physics and Chemistry , 541 (2005).[26] S.T. Epstein, The Variation Method in Quantum Chemistry , (Academic Press, New York(1974)).[27] A.M. Frolov, J. Chem. Phys. , 104302 (2007). ABLE I: The expectation values of a number of electron-nuclear ( en ) and electron-electron ( ee )properties (in a.u. ) of the ground 2 S − and 2 S − states of the of the Li − ( ∞ Li − ) ion and neutralLi ( ∞ Li) atom, respectively.atom/ion state h r − eN i h r − eN i h r eN i h r eN i h r eN i h r eN i Li − S S h r − ee i h r − ee i h r ee i h r ee i h r ee i h r ee i Li − S S E h p e i h p N i h δ eN i h δ ee i h δ eee i Li − S -7.5007605 1.875509 7.809830 3.42829 9.1421 × − S -7.47800737 2.49268725 7.77990315 4.607933 0.181640 0.0 ABLE II: The total energies and some electron-nuclear ( eN ) and electron-electron ( ee ) propertiesin a.u. of a few selected heavy three-electron ions in their the ground 2 S − state(s).ion Sc Ti V Cr E -475.0551425155 -522.4072925498 -572.0094468708 -623.8616048933 h r − eN i h r − ee i h r − eN i h r − ee i h r eN i h r ee i Fe Co Ni E -677.9637661344 -734.3159301916 -792.9180967274 -853.7702654564 h r − eN i h r − ee i h r − eN i h r − ee i h r eN i h r ee i ABLE III: The total non-relativistic energies E of the different atoms/ions in their ground (bound)states in atomic units. All nuclear masses are infinite. Q is the electric charge of the atomic nucleusand N e is the total number of bounded electrons. All energies for the He-like atoms/ions and someenergies of the Li-like ions (after Q = 20) have been determined in this study. This Table is usefulfor accurate eveluations of binding energies, relativistic corrections, etc in few-electron atoms/ions. Q He-like ( N e = 2) Li-like ( N e = 3) Be-like ( N e = 4)1 -0.5277510165443771965925 —————– —————-2 -2.90372437703411959831115924519440 —————– —————-3 -7.27991341266930596491875 -7.4780603236503 —————-4 -13.65556623842358670208051 -14.3247631764654 -14.6673564079515 -22.03097158024278154165469 -23.424605720957 -24.3488843819026 -32.40624660189853031055685 -34.775511275626 -36.5348522852027 -44.781445148772704645183 -48.376898319137 -51.2227126161438 -59.156595122757925558542 -64.228542082701 -68.4115416575899 -75.531712363959491104856 -82.330338097298 -88.10092767635410 -93.906806515037549421417 -102.682231482398 -110.29066107006911 -114.28188377607272189582 -125.2841907536473 -134.98062460425712 -136.65694831264692990427 -150.1361966044594 -162.17074790669213 -161.03200302605835987252 -177.238236559961 -191.86098633826214 -187.40704999866292631487 -206.5903022122780 -224.05131029801215 -215.78209076353716023462 -238.1923876941461 -258.74169942716016 -246.15712647425473932009 -272.0444887900725 -295.93213928864617 -278.53215801540009570337 -308.1466023952556 -335.62261937507518 -312.90718607661114879880 -346.4987261736714 -377.81313186605019 -349.28221120345316700447 -387.1008583345610 -422.50367082665820 -387.65723383315855621790 -429.9529974827626 -469.69423167526521 -428.03225432023469116264 -475.0551425155 -519.38481082107422 -470.40727295513838395930 -522.4072925498 -571.57540541167123 -514.78228997811177388135 -572.0094468708 -626.26601315366224 -561.15730558958127234352 -623.8616048933 -683.45663218292025 -609.53231995807574620568 -677.9637661344 -743.14726096906426 -659.90733322632780520901 -734.3159301916 -805.33789824504027 -712.28234551602655145614 -792.9180967274 -870.02854295168628 -766.65735693155709991040 -853.7702654564 -937.21919419913530 ———————— —————– -1079.10051340709836 ———————— —————– -1564.744568198454= 4)1 -0.5277510165443771965925 —————– —————-2 -2.90372437703411959831115924519440 —————– —————-3 -7.27991341266930596491875 -7.4780603236503 —————-4 -13.65556623842358670208051 -14.3247631764654 -14.6673564079515 -22.03097158024278154165469 -23.424605720957 -24.3488843819026 -32.40624660189853031055685 -34.775511275626 -36.5348522852027 -44.781445148772704645183 -48.376898319137 -51.2227126161438 -59.156595122757925558542 -64.228542082701 -68.4115416575899 -75.531712363959491104856 -82.330338097298 -88.10092767635410 -93.906806515037549421417 -102.682231482398 -110.29066107006911 -114.28188377607272189582 -125.2841907536473 -134.98062460425712 -136.65694831264692990427 -150.1361966044594 -162.17074790669213 -161.03200302605835987252 -177.238236559961 -191.86098633826214 -187.40704999866292631487 -206.5903022122780 -224.05131029801215 -215.78209076353716023462 -238.1923876941461 -258.74169942716016 -246.15712647425473932009 -272.0444887900725 -295.93213928864617 -278.53215801540009570337 -308.1466023952556 -335.62261937507518 -312.90718607661114879880 -346.4987261736714 -377.81313186605019 -349.28221120345316700447 -387.1008583345610 -422.50367082665820 -387.65723383315855621790 -429.9529974827626 -469.69423167526521 -428.03225432023469116264 -475.0551425155 -519.38481082107422 -470.40727295513838395930 -522.4072925498 -571.57540541167123 -514.78228997811177388135 -572.0094468708 -626.26601315366224 -561.15730558958127234352 -623.8616048933 -683.45663218292025 -609.53231995807574620568 -677.9637661344 -743.14726096906426 -659.90733322632780520901 -734.3159301916 -805.33789824504027 -712.28234551602655145614 -792.9180967274 -870.02854295168628 -766.65735693155709991040 -853.7702654564 -937.21919419913530 ———————— —————– -1079.10051340709836 ———————— —————– -1564.744568198454