Calculation of the magnetic hyperfine structure constant of alkali metals and alkaline earth metal ions using the relativistic coupled-cluster method
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Calculation of the magnetic hyperfine structure constant of alkali metals and alkalineearth metal ions using the relativistic coupled-cluster method
Sudip Sasmal ∗ Electronic Structure Theory Group, Physical Chemistry Division,CSIR-National Chemical Laboratory, Pune, 411008, India
The Z-vector method in the relativistic coupled-cluster framework is used to calculate magnetichyperfine structure constant ( A J ) of alkali metals and singly charged alkaline earth metals in theirground state electronic configuration. The Z-vector results are in very good agreement with theexperiment. The A J values of Li, Na, K, Rb, Cs, Be + , Mg + , Ca + , and Sr + obtained in the Z-vectormethod are compared with the extended coupled-cluster results taken from Phys. Rev. A ,022512 (2015). The same basis and cutoff are used for the comparison purpose. The comparisonshows that Z-vector method with the singles and double approximation can produce more precisewavefunction in the nuclear region than the ECC method. PACS numbers: 31.15.aj, 31.15.am, 31.15.bw
I. INTRODUCTION
The interaction of electromagnetic field of electronswith the nuclear moments of nucleus, known as hyper-fine structure interaction, causes small shift and split-ting of energy levels [1]. Therefore, it is very impor-tant for the accurate description of energy levels of atom,molecule and ion. The precise measurements of the en-ergy levels of alkali metals play important role in var-ious areas of atomic and nuclear physics as they areextensively used in high precession spectroscopy, lasercooling and trapping of atom, ultracold collision studies,photo-association spectroscopy, Bose-Einstein condensa-tion and more recently, test for parity and time reversalviolation. Currently, the hyperfine transition of Cs atom[[Xe]6S( S / , F = 3 , m F = 0) ↔ [Xe]6S( S / , F =4 , m F = 0)] is used as frequency standard which is ac-curate up to 1 per 10 [2]. Singly ionized alkaline earthmetal ions are insensitive to the perturbation of the en-vironment arising form collisions and Doppler shift andthus, have been considered for the potential candidatesfor optical frequency standard [3–6]. The S / groundstate of these ions are regarded for quantum informationprocessing studies to encode quantum-bit into hyperfinelevels because of their long phase coherence due to theirsmall energy gap and relatively large spontaneous decaylifetime [7, 8].Recently, experiments for parity non-conservation(PNC) become a cutting-edge topic as it can test the ac-curacy of fundamental physics and explore “new physics”beyond the standard model. However, the PNC ampli-tudes, which are very essential to determine the value ofPNC constants cannot be measured experimentally andthus, has to be obtained theoretically. Therefore, it is ex-tremely important to have a reliable way of determiningthe accuracy of such theoretical calculations. The PNC ∗ [email protected] amplitudes are very sensitive to the accuracy of the wave-function in the near nuclear region [9, 10]. The same istrue for HFS constants [11]. Therefore, one can assess theaccuracy of PNC amplitudes by comparing theoreticallyobtained HFS constants with corresponding experimen-tal values [12–14].Relativistic effects are very important for the precisecalculation of the wavefunction in the near nuclear re-gion. For a single determinant theory, the best way toinclude relativistic effect is to solve the four-componentDirac-Hartree-Fock (DHF) equation. However, the DHFmethod misses the instantaneous interaction of oppositespin electrons. Coupled-cluster (CC) [15–17] is the mostelegant method to include this dynamic electron correla-tion.The coupled-cluster equation can be solved either byvariationally or by non-variationally. Although, the non-variational coupled-cluster, also known as normal cou-pled cluster (NCC) is the most familiar, the variationalcoupled-cluster (VCC) has several advantages over theNCC. The VCC, being variational has upper bounded-ness in energy and satisfies the generalized Hellmann-Feynman (GHF) theorem which simplifies the calcula-tions for higher order properties. Unitary coupled-cluster(UCC) [18–23], expectation value coupled-cluster (XCC)[24–27] and extended coupled-cluster (ECC) [28, 29] arethe most familiar VCC [30] in literature. Recently, ECChas been extended to the relativistic regime to calcu-late magnetic HFS constants of atoms and molecules [31].ECC uses dual space of right and left vectors in a doublelinked form where the left vector is not complex conju-gate of the right vector. Although ECC functional is aterminating series, the natural termination leads to veryexpensive terms. Thus, for practical purpose, one needsto use some truncation scheme to avoid computationallyexpensive terms.On the other hand, the NCC is nonvariational andthus, does not satisfy the GHF theorem. Therefore, theexpectation value and first order energy derivative yielddifferent results [32, 33]. However, the energy derivativemethod is superior than the expectation value method asthe property value obtained in energy derivative methodcan be expressed as the corresponding expectation valueplus some additional terms which make it closer to thefull configuration interaction property value. The NCCenergy is not optimized with respect to determinantalcoefficients ( C d ) in the expansion of many electron wave-function [32]. Thus, the derivative of energy with respectto external perturbation requires the derivative of en-ergy with respect to C d times the derivative of C d withrespect to external perturbation. The derivative termsinvolving C d can be included in Z-vector [34, 35] methodby introducing a perturbation independent de-excitationoperator where the equation for this operator is linear.Thus, for any number of property calculation, one needsto calculate only one set of coupled-cluster amplitudes.The advantage of Z-vector method over ECC method isthat unlike ECC, the equations for excitation operatorsare decoupled from the de-excitation operators. Thissaves enormous computational cost. Recently, Z-vectormethod is extended to the relativistic region for the calcu-lation of ground state properties of atomic and molecularsystems [36].In this paper, we have calculated the magnetic HFSconstant of alkali metal atoms and singly charged alkalineearth metal cations using Z-vector technique in the rel-ativistic coupled-cluster framework. We have comparedthe Z-vector values with the ECC values calculated inRef. [31] to show that the Z-vector method with sin-gle and double approximation can produce much betterwavefunction in the nuclear region of atomic nucleus thanthe ECC method with the approximation stated in thepaper and thus, is capable of providing the precise valueof the types of property like PNC amplitudes, which areprominent in the nuclear region. The paper is organize asfollows. A brief introduction and the workable equationsfor the Z-vector method are given in Sec. II followed bythe matrix elements for the magnetic HFS constant ofatomic system in the same section. The computationaldetails are given in Sec. III. In Sec. IV, we present ourresults and discuss about those before making our finalremarks in Sec. V. II. THEORYA. Z-vector method
The study of hyperfine interaction helps us to under-stand nuclear structure of an atom and its impact on theelectronic wavefunction in the nuclear and near nuclearregion. Therefore, for the accurate calculation of mag-netic HFS constant, which demands very precise wave-function in the short range of nucleus, we need to in-corporate both relativistic and electron correlation ef-fects. In this work, the four component Dirac-Hartree-Fock (DHF) method is used to include the effect of rel-ativity where the electron-electron repulsion term is ap- proximated as Coulomb interaction. The Dirac-CoulombHamiltonian is given by H DC = X i h − c ( ~α · ~ ∇ ) i + ( β − ) c + V nuc ( r i ) + X j>i r ij i , (1)where, α and β are the usual Dirac matrices, c is thespeed of light, is the 4 × V nuc ( r i )is the nuclear potential function and the Gaussian chargedistribution is used in this work. The DHF methodmisses the instantaneous dynamic correlation of oppositespin electrons. Among various many-body theory, thesingle reference coupled-cluster (SRCC) is the most ele-gant technique to incorporate dynamic correlation. TheSRCC wavefunction is given as | Ψ cc i = e T | Φ i , (2)where, Φ is the DHF wavefunction and T is the coupled-cluster excitation operator which is given by T = T + T + · · · + T N = N X n T n , (3)with T m = 1( m !) X ij...ab... t ab...ij... a † a a † b . . . a j a i . (4)Here, i,j(a,b) are the hole(particle) indices and t ab..ij.. arethe cluster amplitudes corresponding to the cluster op-erator T m . In the coupled-cluster single and double(CCSD) approximation, T = T + T . The equationsfor T and T are given as h Φ ai | ( H N e T ) c | Φ i = 0 , h Φ abij | ( H N e T ) c | Φ i = 0 , (5)where, H N is the normal ordered DC Hamiltonian andsubscript c means only the connected terms exist in thecontraction between H N and T. Size-extensivity is en-sured by this connectedness. The coupled-cluster corre-lation energy can be obtained as E corr = h Φ | ( H N e T ) c | Φ i . (6)However, the SRCC energy is not optimized with re-spect to the determinantal coefficients and the molecularorbital coefficients in the expansion of the many electroncorrelated wavefunction [32]. Therefore, the calculationof SRCC energy derivative with respect to external per-turbation requires to include these derivative terms. Theequation for these terms are linear but in general, per-turbation dependent. However, in Z-vector method, thederivative terms containing the determinantal coefficientscan be incorporated by the introduction of a perturba-tion independent operator Λ [35]. Thus, in the Z-vectormethod, any number of property calculations can be doneby solving only one set of T and Λ amplitudes. Λ is adeexcitation operator and the second quantized form isgiven by Λ = Λ + Λ + ... + Λ N = N X n Λ n , (7)where, Λ m = 1( m !) X ij..ab.. λ ij..ab.. a † i a † j ....a b a a . (8)Here λ ij..ab.. are the cluster amplitudes corresponding tothe cluster operator Λ m . In CCSD approximation, Λ =Λ + Λ . The explicit equations for the amplitudes of Λ and Λ operators are given by h Φ | [Λ( H N e T ) c ] c | Φ ai i + h Φ | ( H N e T ) c | Φ ai i = 0 , (9) h Φ | [Λ( H N e T ) c ] c | Φ abij i + h Φ | ( H N e T ) c | Φ abij i + h Φ | ( H N e T ) c | Φ ai ih Φ ai | Λ | Φ abij i = 0 . (10)The energy derivative is given by∆ E ′ = h Φ | ( O N e T ) c | Φ i + h Φ | [Λ( O N e T ) c ] c | Φ i . (11)Here, O N is the derivative of normal ordered perturbedHamiltonian with respect to external field of perturba-tion. It is clear from the above formulation that thederivative terms containing only the determinantal co-efficients are included here, i.e., the orbital relaxationterms that are required to make energy functional sta-tionary with respect to molecular orbital coefficients arenot considered here. It is worth to mention that recently,Saue and coworkers [37] have implemented the orbital-unrelaxed analytical method in the four-component rela-tivistic SRCC framework based on the Lagrangian mul-tiplier method of Helgaker and coworkers [38] which issimilar to the Z-vector method for the ground state firstorder properties. B. Magnetic hyperfine structure constant
The magnetic HFS interaction arises due to the cou-pling of nuclear magnetic moment with the angular mo-mentum of electrons and thus, can be treated as a one-body interaction from the electronic structure point ofview. The magnetic vector potential due to a nucleus isgiven by ~A = ~µ k × ~rr , (12)where, ~µ k is the magnetic moment of nucleus K . In Diractheory, the HFS interaction Hamiltonian due to ~A can begiven as H hfs = n X i ~α i · ~A i , (13) where, α i denotes the Dirac α matrices for the i th electronand n is the total no of electrons. The magnetic hyperfineconstant of the J th electronic state of an atom can begiven as A J = 1 IJ h Ψ J | H hfs | Ψ J i = ~µ k IJ · h Ψ J | n X i (cid:18) ~α i × ~r i r i (cid:19) | Ψ J i , (14)where, I is the nuclear spin quantum number and Ψ J isthe wavefunction of the J th electronic state. III. COMPUTATIONAL DETAILS
TABLE I: Basis and cutoff used for the atomic calculation.Atom Basis Virtual cutoff (a.u.)Li aug-cc-pCVQZNa aug-cc-pCVQZK dyall.cv4z 500Rb dyall.cv3z 500Cs dyall.cv4z 40Fr dyall.cv3z 50Be + aug-cc-pCVQZMg + aug-cc-pCVQZCa + dyall.cv4z 500Sr + dyall.cv3z 100Ba + dyall.cv4z 40Ra + dyall.cv3z 50 The DIRAC10 program package [39] is used to solvethe DHF equation and to construct the one-electron andtwo-electron matrix elements. The magnetic HFS in-tegrals are extracted from a locally modified version ofDIRAC10. Gaussian charge distribution is considered forthe finite size of the nucleus where the the nuclear param-eters are taken from Ref. [40]. Restricted kinetic balance[41] condition is used to link small and large componentbasis function. No virtual pair approximation (NVPA)is used to solve DHF equation. This means that the neg-ative energy solutions are removed by using projectionoperator and only positive energy solutions are includedin the correlation calculations. However, how to go be-yond the no-pair approximation by accounting for cor-relation contributions of negative energy states has beendiscussed in depth in Ref. [42–44]. In our calculation,we have used aug-cc-pCVQZ basis [45, 46] for Li, Na, Beand Mg atoms and dyall.cv3z basis [47] for Rb, Sr, Fr andRa atoms and dyall.cv4z [47] basis for K, Ca, Cs and Baatoms. All electrons are considered for the correlationcalculation of all systems. The cutoff used for the virtualorbitals are compiled in Table I.
IV. RESULTS AND DISCUSSION
TABLE II: Magnetic hyperfine coupling constant (in MHz) ofground state of atoms.
Atom SCF ECC Z-vector Expt. δ %[31] ECC Z-vector Li 107.2 149.3 148.3 152.1 [48] 1.9 2.6 Li 283.2 394.3 391.6 401.7 [48] 1.9 2.6 Na 630.6 861.8 861.4 885.8 [48] 2.8 2.8 K 151.0 223.5 226.6 230.8 [48] 3.3 1.9 K -187.7 -277.9 -281.8 -285.7 [49] 2.8 1.4 K 82.9 122.7 124.4 127.0 [48] 3.5 2.1 Rb 666.9 972.5 986.5 1011.9 [50] 4.1 2.6 Rb 2260.1 3295.7 3343.3 3417.3 [51] 3.7 2.2
Cs 1495.5 2179.1 2218.4 2298.1 [52] 5.5 3.6
Fr 5518.0 7537.4 7654(2)[53] 1.5 Be + -498.8 -614.6 -612.9 -625.0 [54] 1.7 2.0 Mg + -466.7 -581.6 -584.8 -596.2 [55] 2.5 1.9 Ca + -606.2 -794.9 -801.5 -806.4 [56] 1.4 0.6 Sr + -761.0 -969.9 -977.9 -1000.5(1.0) [57] 3.2 2.3 Ba + Ba + Ra + In Table II, we present the magnetic HFS constantof alkali metal atoms and mono-positive alkaline earthmetal ions in their ground state ( S ) electronic config-uration using the Z-vector technique in the relativisticcoupled-cluster framework. We have compiled the exper-imental values for these systems in the same table andthe relative deviations of Z-vector results from the exper-imental values are presented as δ %. The results for differ-ent isotopes are calculated by using their correspondingnuclear magnetic moment values although the nuclearparameters in the nuclear model are taken as same foreach isotope which is default the most stable isotopesin DIRAC10 [39]. Our calculated Z-vector results are invery good agreement with the experimental values. Fromthe table, it is clear that the deviations of Z-vector resultsfrom the experiments are well within 3% except for the Cs atom, where it is 3.6%. The Z-vector results arequite impressive, especially for the heavy atoms. TheECC values of magnetic HFS constant are taken fromRef. [31] and the deviations from the experiment arepresented in the table. We have used same basis andcutoff for those systems for comparison purpose. Thedeviations of ECC and Z-vector values from the experi-mental values are presented in Fig. 1. From the figure, itis clear that Z-vector results are far better than the ECCresults except for two small systems like Li and Be + . Asthe magnetic HFS constant is very sensitive to the nearnuclear wavefunction, the above results show that the Z-vector method can produce far better wavefunction in thenuclear region than the ECC method and the results arequite impressive for the heavy atoms. Although, ECCis a truncated series, in CCSD model, the natural trun-cation leads to very expensive terms. In Ref. [31], thetruncation scheme proposed by Vaval et al are used toavoid the expensive terms in the ECC functional where FIG. 1: Comparison of relative deviations between Z-vectorand ECC values of magnetic HFS constant of atoms. the right exponential is full within the CCSD approxi-mation and the higher-order double-linked terms withinthe CCSD approximation are taken in the left exponent.This approximation introduces an additional error whichmay be the reason for the poor performance of ECC com-pared to Z-vector method.
TABLE III: Comparison of full CI and Z-vector magneticHFS values (in MHz) of LiBasis Full CI [31] Z-vectoraug-cc-pCVDZ 384.1 383.9aug-cc-pCVTZ 402.0 401.3aug-cc-pCVQZ a a Considering 3 electrons and 189 virtual orbitalsTABLE IV: Comparison of full CI and Z-vector magnetic HFSvalues (in MHz) of Be + Basis Full CI [31] Z-vectoraug-cc-pCVDZ -586.6 -586.5aug-cc-pCVTZ -615.7 -615.4aug-cc-pCVQZ a -613.0 -612.7 a Considering 3 electrons and 183 virtual orbitals
The HFS constant predominantly depends on the spindensity of the valence electron in the nuclear region andthus is not very sensitive to the retardation and magneticeffects described by the Breit interaction [61, 62]. It canbe seen from the previous calculations that the higherorder relativistic effects on these types of properties gen-erally lie ∼ Li and Be + has been made and is presented inTables III and IV, respectively. By comparing Z-vectorvalues with FCI values and considering all other sourcesof error like higher order relativistic effects, missing cor-relation effects etc, it can be assumed that the overalluncertainty in our final results is less than 5%. V. CONCLUSION
We have calculated the magnetic HFS constant of al-kali metal atoms (Li, Na, K, Rb, Cs and Fr) and mono-positive alkaline earth metal ions (Be + , Mg + , Ca + , Sr + ,Ba + and Ra + ) using the Z-vector technique in the rel-ativistic coupled-cluster framework. We have compared the Z-vector values and the ECC values taken from Ref.[31] with experiment and the comparison shows that theZ-vector method with single and double approximationcan produce much accurate wavefunction in the nuclearregion than the ECC method with the given approxima-tion. A possible explanation for the poor performance ofthe ECC method is also given. Acknowledgement
Author thanks Prof. Sourav Pal, Dr. Malaya K.Nayak, Dr. Nayana Vaval, Dr. Himadri Pathak for theirvaluable comments and insights in this work. Authoracknowledges Dr. Malaya K. Nayak for providing thehyperfine integrals. Author acknowledges the resourcesof the Center of Excellence in Scientific Computing atCSIR-NCL. S.S. acknowledges the CSIR for the SPM fel-lowship. [1] I. Lindgren and J. Morrison,
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