Theory and applications of atomic and ionic polarizabilities
Atomic polarization phenomena impinge upon a number of areas and processes in physics. The dielectric constant and refractive index of any gas are examples of macroscopic properties that are largely determined by the dipole polarizability. When it comes to microscopic phenomena, the existence of alkaline-earth anions and the recently discovered ability of positrons to bind to many atoms are predominantly due to the polarization interaction. An imperfect knowledge of atomic polarizabilities is presently looming as the largest source of uncertainty in the new generation of optical frequency standards. Accurate polarizabilities for the group I and II atoms and ions of the periodic table have recently become available by a variety of techniques. These include refined many-body perturbation theory and coupled-cluster calculations sometimes combined with precise experimental data for selected transitions, microwave spectroscopy of Rydberg atoms and ions, refractive index measurements in microwave cavities, ab initio calculations of atomic structures using explicitly correlated wave functions, interferometry with atom beams, and velocity changes of laser cooled atoms induced by an electric field. This review examines existing theoretical methods of determining atomic and ionic polarizabilities, and discusses their relevance to various applications with particular emphasis on cold-atom physics and the metrology of atomic frequency standards.
pacs:31.15.ap, 32.10.Dk, 42.50.Hz, 51.70.+f
By the time Maxwell presented his article on a Dynamical Theory of the Electromagnetic Field Maxwell (1864), it was understood that bulk matter had a composition of particles of opposite electrical charge, and that an applied electric field would rearrange the distribution of those charges in an ordinary object. This rearrangement could be described accurately even without a detailed microscopic understanding of matter. For example, if a perfectly conducting sphere of radius is placed in a uniform electric field , simple potential theory shows that the resulting electric field at a position outside the sphere must be . This is equivalent to replacing the sphere with a point electric dipole,
where is the dipole polarizability of the sphere111For notational convenience, we use the Gaussian system of electrical units, as discussed in subsection A below. In the Gaussian system, electric polarizability has the dimensions of volume.. An arbitrary applied electric field can be decomposed into multipole fields of the form , where is a spherical tensor Edmonds (1996). Each of these will induce a multipole moment of in the conducting sphere, corresponding to a multipole polarizability of Treatment of the electrical polarizabilities of macroscopic bodies is a standard topic of textbooks on electromagnetic theory, and the only material properties that it requires are dielectric constants and conductivities.
Quantum mechanics, on the other hand, offers a fundamental description of matter, incorporating the effects of electric and magnetic fields on its elementary constituents, and thus enables polarizabilities to be calculated from first principles. The standard framework for such calculations, perturbation theory, was first laid out by Schrödinger Schrödinger (1926) in a paper that reported his calculations of the Stark effect in atomic hydrogen. A system of particles with positions and electric charges exposed to a uniform electric field, , is described by the Hamiltonian
where is the Hamiltonian in the absence of the field, and is the dipole moment operator,
Treating the field strength, , as a perturbation parameter, means that the energy and wave function can be expanded as
The first-order energy if is an eigenfunction of the parity operator. In this case, satisfies the equation
From the solution to Eq. (6), we can find the expectation value
where is a matrix. The second-order energy is given by
Although Eq. (6) can be solved directly, and in some cases in closed form, it is often more practical to express the solution in terms of the eigenfunctions and eigenvalues of , so that Eq. (8) takes the form
This sum over all stationary states shows that calculation of atomic polarizabilities is a demanding special case of the calculation of atomic structure. The sum extends in principle over the continuous spectrum, which sometimes makes substantial contributions to the polarizability.
|Table||System||Atoms and Ions||States||Data|
|Table 4||Noble gases||He, Ne, Ar, Kr, Xe, Rn, Li, Na, K, Rb, Cs, Fr||ground|
|Be, Mg, Ca, Sr, Ba, Ra|
|Table 5||Alkali atoms||Li, Na, K, Rb, Cs, Fr||ground ,||,|
|Table 6||Alkali ions||Be, Mg, Ca, Sr, Ba, Ra||ground|
|Table 8||Monovalent||Li, Na, K, Ca, Rb, Sr||excited||,|
|Table 7||Alkali atoms||Resonance transition: Li, Na, K, Rb, Cs||,|
|Table 9||Alkali atom||Na||ground|
|Table 10||Alkali atom||Cs||26 states||,|
|Table 11||Group II type||Be, Mg, Ca, Sr, Ba, Ra, Al, Si, Zn, Cd, Hg, Yb||ground,|
|Table 12||Miscellaneous||Al, Ga, In, Tl, Si, Sn, Pb, Ir, U, Cu, Ag, Au,||ground|
|Al, Si, P, Kr, Cu, Ag, Hg, Yb, Zn|
|Table 13||Miscellaneous||Ca, Sr, Ba, Zn, Cd, Hg, Yb, Al, Tl, Yb||excited||,|
|Table 14||Miscellaneous||Li, Na, Cs, Mg, Ca, Ba, Hg, Ga, Tl, Yb|
|Table 16||Miscellaneous||Mg, Ca, Sr, Yb, Zn, Cd, Hg,||clock|
|Ca, Sr, Hg, Yb, Al, In||transition|
|Table 18||Monovalent||Li, Na, K, Rb, Cs, Ba, Yb, Hg||ground hyperfine||BBR|
|Table 19||Alkali atoms||Li, Na, K, Rb, Cs, Fr||ground|
Interest in the subject of polarizabilities of atomic states has recently been elevated by the appreciation that the accuracy of next-generation atomic time and frequency standards, based on optical transitions Madej and Bernard (2001); Udem et al. (2002); Diddams et al. (2004); Gill et al. (2003); Gill (2005); Margolis (2009), is significantly limited by the displacement of atomic energy levels due to universal ambient thermal fluctuations of the electromagnetic field: blackbody radiation (BBR) shifts Gallagher and Cooke (1979); Itano et al. (1982); Hollberg and Hall (1984); Porsev and Derevianko (2006a). This phenomenon brings the most promising approach to a more accurate definition of the unit of time, the second, into contact with deep understanding of the thermodynamics of the electromagnetic radiation field.
Description of the interplay between these two fundamental phenomena is a major focus of this review, which in earlier times might have seemed a pedestrian discourse on atomic polarizabilities. The precise calculation of atomic polarizabilities also has implications for quantum information processing and optical cooling and trapping schemes. Modern requirements for precision and accuracy have elicited renewed attention to methods of accurate first-principles calculations of atomic structure, which recently have been increased in scope and precision by developments in methodology, algorithms, and raw computational power. It is expected that the future will lead to an increased reliance on theoretical treatments to describe the details of atomic polarization. Indeed, at the present time, many of the best estimates of atomic polarizabilities are derived from a composite analysis which integrates experimental measurements with first principles calculations of atomic properties.
There have been a number of reviews and tabulations of atomic and ionic polarizabilities Dalgarno (1962); Teachout and Pack (1971); Miller and Bederson (1977); Miller (1988); van Wijngaarden W A (1996); Bonin and Kresin (1997); Delone and Krainov (1999); Schwerdtfeger (2006); Gould and Miller (2005); Lundeen (2005); Miller (2007); Lupinetti and Thakkar (2006). Some of these reviews, e.g. Miller and Bederson (1977); Miller (1988); Gould and Miller (2005); Lundeen (2005) have largely focussed upon experimental developments while others Bonin and Kresin (1997); Schwerdtfeger (2006); Lupinetti and Thakkar (2006) have given theory more attention.
In the present review, the strengths and limitations of different theoretical techniques are discussed in detail given their expected importance in the future. Discussion of experimental work is mainly confined to presenting a compilation of existing results and very brief overviews of the various methods. The exception to this is the interpretation of resonance excitation Stark ionization spectroscopy Lundeen (2005) since issues pertaining to the convergence of the perturbation analysis of the polarization interaction are important here. The present review is confined to discussing the polarizabilities of low lying atomic and ionic states despite the existence of a body of research on Rydberg states Gallagher (2005). High-order polarizabilities are not considered except in those circumstances where they are specifically relevant to ordinary polarization phenomenon. The influence of external electric fields on energy levels comprises part of this review as does the nature of the polarization interaction between charged particles with atoms and ions. The focus of this review is on developments related to contemporary topics such as development of optical frequency standards, quantum computing, and study of fundamental symmetries. Major emphasis of this review is to provide critically evaluated data on atomic polarizabilities. Table 1 summarizes the data presented in this review to facilitate the search for particular information.
i.1 Systems of units
Dipole polarizabilities are given in a variety of units, depending on the context in which they are determined. The most widely used unit for theoretical atomic physics is atomic units (a.u.), in which, , , and the reduced Planck constant have the numerical value . The polarizability in a.u. has the dimension of volume, and its numerical values presented here are thus expressed in units of , where nm is the Bohr radius. The preferred unit systems for polarizabilities determined by experiment are , kHz/(kV/cm), cm/mol or Cm/N where Cm/N is the SI unit. In this review, almost all polarizabilities are given in a.u. with uncertainties in the last digits (if appropriate) given in parentheses. Conversion factors between the different units are listed in Table 2. The last line of the table gives conversion factors from SI units to the other units. For example, the atomic units for can be converted to SI units by multiplying by 0.248832.
Stark shift experiments which measure the change in photon frequency of an atomic transition as a function of electric field strength are usually reported as a Stark shift coefficient in units of kHz/(kV/cm). The polarizability difference is twice the size of the Stark shift coefficient, as in equation (8).
|Conversion from SI||1|
Ii Atomic Polarizabilities and field-atom interactions
ii.1 Static electric polarizabilities
ii.1.1 Definitions of scalar and tensor polarizabilities
The overall change in energy of the atom can be evaluated within the framework of second-order perturbation theory. Upon reduction, the perturbation theory expression given by Eq.(9) leads to a sum-over-states formula for the static scalar electric-dipole polarizability which is expressed most compactly in terms of oscillator strengths as
In this expression, is the absorption oscillator strength for a dipole transition from level to level , defined in a -representation as Fuhr and Wiese (1995)
where and is the spherical tensor of rank 1 Edmonds (1996). The definition of the oscillator strength in coupling is transparently obtained from Eq. (11) by replacing the total angular momentum by the orbital angular momentum.
The polarizability for a state with non-zero angular momentum depends on the magnetic projection :
The quantity is called the scalar polarizability while is the tensor polarizability in J representation.
The scalar part of the polarizability can be determined using Eq. (10). In terms of the reduced matrix elements of the electric-dipole operator, the scalar polarizability of an atom in a state with total angular momentum and energy is also written as
The tensor polarizability is defined as
It is useful in some cases to calculate polarizabilities in strict coupling. In such cases Mitroy and Bromley (2004), the tensor polarizability for a state with orbital angular momentum L is given by
The tensor polarizabilities and in the and representations, respectively, are related by
Equation (14) indicates that spherically symmetric levels (such as the and levels of cesium) only have a scalar polarizability. However, the hyperfine states of these levels can have polarizabilities that depend upon the hyperfine quantum numbers and . The relationship between and polarizabilities is discussed in Ref. Arora et al. (2007). This issue is discussed in more detail in the section on BBR shifts.
There are two distinctly different broad approaches to the calculation of atomic polarizabilities. The “sum-over-states” approach uses a straightforward interpretation of Eq. (9) with the contribution from each state being determined individually, either from a first principles calculation or from interpretation of experimental data. A second class of approaches solves inhomogeneous equation (6) directly. We refer to this class of approaches as direct methods, but note that there are many different implementations of this strategy.
ii.1.2 The sum-over-states method
The sum-over-states method utilizes expression such as Eqs. (10, 13 - 15) to determine the polarizability. This approach is widely used for systems with one or two valence electrons since the polarizability is often dominated by transitions to a few low lying excited states. The sum-over-states approach can be used with oscillator strengths (or electric-dipole matrix elements) derived from experiment or atomic structure calculations. It is also possible to insert high-precision experimental values of these quantities into an otherwise theoretical determination of the total polarizability. For such monovalent or divalent systems, it is computationally feasible to explicitly construct a set of intermediate states that is effectively complete. Such an approach is computationally more difficult to apply for atoms near the right hand side of the periodic table since the larger dimensions involved would preclude an explicit computation of the entire set of intermediate state wave functions.
For monovalent atoms, it is convenient to separate the total polarizability of an atom into the core polarizability and the valence part defined by Eq. (13). The core contribution actually has two components, the polarizability of the ionic core and a small change due to the presence of the valence electron Arora et al. (2007a). For the alkali atoms, the valence part of the ground state polarizability is completely dominated by the contribution from the lowest excited state. For example, the and transitions contribute more than 99% of the Rb valence polarizability Safronova et al. (1999). The Rb core polarizability contributes about 3%. Therefore, precision experimental measurements of the transition rates for the dominant transitions can also be used to deduce accurate values of the ground state polarizability. However, this is not the case for some excited states where several transitions may have large contributions and continuum contributions may be not negligible.
This issue is illustrated using the polarizability of the state of the Rb atom Arora et al. (2007a), which is given by
We present a solution to the Eq. (II.1.2) that combines first principles calculations with experimental data. The strategy to produce a high-quality recommended value with this approach is to calculate as many terms as realistic or feasible using the high-precision atomic structure methods. Where experimental high-precision data are available (for example, for the transitions) they are used in place of theory, assuming that the expected theory uncertainty is higher than that of the experimental values. The remainder that contains contributions from highly-excited states is generally evaluated using (Dirac-Hartree-Fock) DHF or random-phase approximation (RPA) methods. In our example, the contribution from the very high discrete () and continuum states is about 1.5% and cannot be omitted in a precision calculation. Table 3 lists the dipole matrix elements and energy differences required for evaluation of Eq. (II.1.2) as well as the individual contributions to the polarizability. Experimental values from Volz and Schmoranzer (1996) are used for the matrix elements, otherwise the matrix elements are obtained from the all-order calculations of Ref. Arora et al. (2007a) described in Section IV.6. Absolute values of the matrix elements are given. Experimental energies from Sansonetti et al. (2005); Moore (1971) are used. Several transitions give significant contributions. This theoretical number agrees with experimental measurement within the uncertainty. The comparison with experiment is discussed in Section V.
ii.1.3 Direct methods
From a conceptual viewpoint, the finite-field method represents one of the simplest ways to compute the polarizability. In this approach, one solves the Schrödinger equation using standard techniques for the perturbed Hamiltonian given by Eq. (2) for a variety of values of . The polarizability is then extracted from the dipole moment or the energy eigenvalues of the perturbed Hamiltonian. This usually entails doing a number of calculations at different discrete field strengths. This approach is generally used to obtain polarizabilities in coupled-cluster calculations (see, for example, Refs. Lim and Schwerdfeger (2004); Lupinetti and Thakkar (2005)). We note that linearized coupled-cluster calculations are implemented very differently, and sum-over-states is used for the polarizability calculations Safronova and Johnson (2008). These differences between coupled-cluster calculations are discussed in Section IV.
Another direct approach to calculating polarizability is the perturbation-variation method Hibbert et al. (1977). The perturbation-variation approach has been outlined in the introduction as Eqs. (5) to (7). The unperturbed state, and perturbed state, would be written as a linear combinations of basis states. Equations (6) and (7) then reduce to sets of matrix equations. A general technique for solving the inhomogeneous equation (6) has been described by Dalgarno and Lewis in Ref. Dalgarno and Lewis (1955).
Exact solutions to Eqs. (5) - (7) are possible for atomic hydrogen and hydrogenic ions. The non-relativistic solutions were first obtained independently in 1926 by Epstein Epstein (1926), Waller Waller (1926), and Wentzel Wentzel (1926); the relativistic case remains a subject of current research interest Yakhontov (2003); Szmytkowski and Mielewczyk (2004); Szmytkowski (2006); Jentschura and Haas (2008). The nonrelativistic equations are separable in parabolic coordinates, and the polarizability of a hydrogenic ion of nuclear charge Z in the state is (in a.u.)
where are parabolic quantum numbers Bethe and Salpeter (1977), is the projection of the orbital angular momentum onto the direction of the electric field, and is the principal quantum number. A convenient special case is , which corresponds to the familiar circular states of hydrogen in spherical coordinates, with orbital angular momentum ; for these states, .
For the H 1 ground state exposed to an electric field , the solution to Eq. (6) is (in a.u.)
from which . Note that although of Eq. (20) is a state, it is much more compact than any of the discrete eigenstates of H. Thus building up by the sum-over-states approach requires a significant contribution from the continuous spectrum of H. This is depicted in Fig. 1, which employs the histogram construction of Fano and Cooper Fano and Cooper (1968) to show the connection between discrete and continuum contributions to the sum over states. About 20% of the polarizability of H 1 comes from the continuum.
Clearly, the direct solution of the Schrödinger equation for an atom in the presence of an electric field and subsequent determination of the polarizability is formally equivalent to the sum-over-states approach described in the previous subsection. However, it is useful to comment on how this equivalence is actually seen in calculations for many-electron atoms. For example, random-phase-approximation (RPA) results for polarizabilities of closed-shell atoms Johnson et al. (1983) that were obtained by direct solution of inhomogeneous equation are the same (up to numerical uncertainty of the calculations) as sum-over-state RPA results obtained using formula
where is reduced matrix element of dipole operator obtained in the DHF approximation and the matrix elements include RPA terms using many-body perturbation theory as discussed, for example, in Johnson et al. (1996). The index refers to all core orbitals, while the includes all other orbitals. The sum-over-states can be calculated with a finite basis set Johnson et al. (1988), and such an approach intrinsically includes the continuum states when complete sum over the entire basis set is carried out. When the contributions from highly-excited states are significant, it becomes difficult to account for these terms accurately within the framework of the sum-over-states approach. Direct method automatically accounts for these states and this problem does not arise. However, it becomes difficult and cumbersome to include corrections to the dipole operator beyond RPA. The method implemented in Johnson et al. (1983) is different from the finite field approach and does not involve performing a number of calculations at different discrete field strengths.
In most high-precision calculations, the determination of polarizabilities follows the calculation of wave functions or quantities that represent the wave functions (such as excitation coefficients). The type of approach used for this initial calculation generally determines whether polarizabilities are determined by Eqs. (6) or by sum-over-states method. For example, relativistic linearized coupled-cluster approach Safronova and Johnson (2008) is formulated in a way that does not explicitly generate numerical wave functions on a radial grid, and all quantities are expressed in terms of excitation coefficients. Therefore, the polarizabilities are calculated by the sum-over-states method using resulting high-quality dipole matrix elements and energies. In the case of methods that combine relativistic configuration interaction and perturbation theory [CI+MBPT], it is natural to determine polarizabilities by directly solving the inhomogeneous equation. In this case, it is solved in the valence space with the effective operators that are determined using MBPT Kozlov and Porsev (1999). The ionic core polarizability is calculated separately in this approach. The effective dipole operator generally includes RPA corrections, with other corrections calculated independently.
The direct and sum-over-states approaches can also be merged in a hybrid approach. One strategy is to perform a direct calculation using the best available techniques, and then replace the transition matrix elements for the most important low-lying states with those from a higher level theory. This hybrid method is discussed further in the sections on the CI+MBPT and CI+all-order methods.
ii.2 The frequency-dependent polarizability
So far, we have described the polarizability for static fields. The numerical value of the polarizability changes when the atom is immersed in an alternating (AC) electromagnetic field. To second order, one writes . The valence part of the scalar frequency dependent polarizability, usually called the dynamic polarizability, is calculated using the sum-over-states approach with a straightforwardly modified version of Eq. (13):
Eq. (22) assumes that is at least a few linewidths away from resonant frequencies defined by . As noted previously, atomic units are used throughout this paper, and . The core part of the polarizability may also be corrected for frequency dependence in random phase approximation by similarly modifying the formula (21). Static values may be used for the core contribution in many applications since the frequencies of interest, (i.e. corresponding to commonly used lasers) are very far from the excitation energies of the core states. The calculations of the ground and excited state frequency-dependent polarizabilities of the alkali-metal atoms are described in detail in Refs. Safronova et al. (2006) and Arora et al. (2007), respectively. It is essentially the same as the calculation of the static polarizability described in Section II.1.2, only for .
The expression for the tensor polarizability given by Eq. (14) is modified in the same way, i.e. by replacing
There has been more interest recently in the determination of frequency-dependent polarizabilities due to the need to know various “magic wavelengths” Katori et al. (1999) for the development of optical frequency standards and other applications. At such wavelengths, the frequency-dependent polarizabilities of two states are the same, and the AC Stark shift of the transition frequency between these two states is zero. An example of the calculation of frequency-dependent polarizabilities and magic wavelengths is given in Section VII.2. Experimentally determined magic wavelengths may also be used to gauge the accuracy of the theory.
Iii Measurements of polarizabilities and related quantities
Experimental measurements of atomic and ionic polarizabilities are somewhat rarer than theoretical determinations. There are two types of measurements, those which directly determine the polarizability, and those which determine differences in polarizabilities of two states from Stark shift of atomic transitions.
For the most part, we make brief comments on the major experimental techniques and refer the reader to primarily experimental reviews Miller (1988); Bonin and Kresin (1997); Gould and Miller (2005); Lundeen (2005) for further details.
iii.1 -sum rules
This approach makes use of Eqs. (10-15). Many of the most interesting atoms used in cold atom physics typically have only one or two valence electrons. The ground state polarizability of these atoms is dominated by a single low-lying transition. As mentioned in Section II.1.2, 97% of the total value of Rb ground state polarizability comes from transition. In the case of Na, about 99.4 of the valence polarizability and 98.8 of the total polarizability of sodium arises from the resonant transition.
Composite estimates of the polarizability using both experimental and theoretical inputs are possible. One type of estimate would use experimental oscillator strengths to determine the valence polarizability. This could be combined with a core contribution obtained by other methods to estimate the total polarizability. Another approach replaces the most important matrix elements in a first-principles calculation by high precision experimental values Derevianko et al. (1999); Porsev and Derevianko (2006b). Various types of experiments may be used to determine particular matrix elements, including photo-association experiments Bouloufa et al. (2009), lifetime, oscillator strengths, or Stark shift measurements Arora et al. (2007a) with photoassociation experiments generally giving the most reliable matrix elements. This hybrid method may provide values accurate to better than 0.5% in certain cases Derevianko et al. (1999).
iii.2 Dielectric constant
The dielectric constant of an atomic or molecular gas is related to the dipole polarizability, , by the identity
where is the atomic number density. The technique has only been applied to the rare gas atoms, and the nitrogen and oxygen atoms by the use of a shock tube. Results for the rare gases typically achieve precisions of . Examples are reported in Table 4.
iii.3 Refractive index
The frequency-dependent refractive index of a gas , is related to the polarizability by the expression
where is the atomic number density. The static dipole polarizability, , can be extracted from the frequency-dependent polarizability by the following technique.
The energy denominator in Eq. (22) can be expanded when the frequency is smaller than the frequency of the first excitation giving
The factors are the Cauchy moments of the oscillator strength distribution and are defined by
Specific Cauchy moments arise in a number of atomic physics applications, as reviewed by Fano and Cooper Fano and Cooper (1968). For example, the Thomas-Reiche-Kuhn sum rule states that is equal to the number of electrons in the atom. The moment is related to the non-adiabatic dipole polarizability Kleinman et al. (1968); Dalgarno et al. (1968).
The general functional dependence of the polarizability at low frequencies is given by Eq. (26) Dalgarno and Kingston (1960); Langhoff and Karplus (1969). The achievable precision for the rare gases is 0.1 or better Langhoff and Karplus (1969); Schmidt et al. (2007). Experiments on the vapours of Zn, Cd and Hg gave polarizabilities with uncertainties of Goebel and Hohm (1995, 1996).
iii.4 Deflection of an atom beam by electric fields
The beam deflection experiment is conceptually simple. A collimated atomic beam is directed through an interaction region containing an inhomogeneous electric field. While the atom is in the interaction region, the electric field induces a dipole moment on the atom. Since the field is not uniform, a force proportional to the gradient of the electric field and the induced dipole moment results in the deflection of the atomic beam. The polarizability is deduced from the deflection of the beam. The overall uncertainty in the derived polarizabilities is between Hall and Zorn (1974). Therefore, this method is mainly useful at the present time for polarizability measurements in atoms inaccessible by any other means.
iii.5 The - balance method
In this approach, the - balance configuration applies an inhomogeneous electric field and an inhomogeneous magnetic field in the interaction region Molof et al. (1974). The magnetic field acts on the magnetic moment of the atom giving a magnetic deflection force in addition to the electric deflection. The experiment is tuned so that the electric and magnetic forces are in balance. The polarizability can be determined since the magnetic moments of many atoms are known. Uncertainties range from to Molof et al. (1974); Miller and Bederson (1976); Schwartz et al. (1974).
iii.6 Atom interferometry
The interferometry approach splits the beam of atoms so that one path sends a beam through a parallel plate capacitor while the other goes through a field free region. An interference pattern is then measured when the beams are subsequently merged and detected. The polarizability is deduced from the phase shift of the beam passing through the field free region. So far, this approach has been used to measure the polarizabilities of helium (see Cronin et al. (2009) for a discussion of this measurement), lithium Miffre et al. (2006), sodium Ekstrom et al. (1995); Holmgren et al. (2010), potassium Holmgren et al. (2010), and rubidium Holmgren et al. (2010) achieving uncertainties of .
It has been suggested that multi-species interferometers could possibly determine the polarizability ratio to relative accuracy Cronin et al. (2009). Consequently, a measurement of in conjunction with a known standard, say lithium, could lead to a new level of precision in polarizability measurements. Already the Na:K and Na:Rb polarizability ratios have been measured with a precision of 0.3 Holmgren et al. (2010).
iii.7 Cold atom velocity change
The experiment of Amini and Gould Amini and Gould (2003) measured the kinetic energy gained as cold cesium atoms were launched from a magneto-optical trap into a region with a finite electric field. The kinetic energy gained only depends on the final value of the electric field. The experimental arrangement actually measures the time of return for cesium atoms to fall back after they are launched into a region between a set of parallel electric-field plates. The only such experiment reported so far gave a very precise estimate of the Cs ground state polarizability, namely a.u.. This approach can in principle be applied to measure the polarizability of many other atoms with a precision approaching 0.1 Gould and Miller (2005).
iii.8 Other approaches
The deflection of an atomic beam by pulsed lasers has been used to obtain the dynamic polarizabilities of rubidium and uranium Kadar-Kallen and Bonin (1992, 1994). The dynamic polarizabilities of some metal atoms sourced from an exploding wire have been measured interferometrically Hu and Kusse (2002); Sarkisov et al. (2006). These approaches measure polarizabilities to an accuracy of 5-20.
iii.9 Spectral analysis for ion polarizabilities
The polarizability of an ion can in principle be extracted from the energies of non-penetrating Rydberg series of the parent systemBorn and Heisenberg (1924); Waller (1926); Mayer and Mayer (1933). The polarizability of the ionic core leads to a shift in the energy levels away from their hydrogenic values.
Consider a charged particle interacting with an atom or ion at large distances. To zeroth order, the interaction potential between a highly excited electron and the residual ion is just
where is the nuclear charge and is the number of electrons. However, the outer electron perturbs the atomic charge distribution. This polarization of the electron charge cloud leads to an attractive polarization potential between the external electron and the atom. The Coulomb interaction in a multipole expansion with , is written as
Applying second-order perturbation theory leads to the adiabatic polarization potential between the charged particle and the atom, e.g.
The quantities are the multipole polarizabilities defined as
In this notation, the electric-dipole polarizability is written as , and is the absorption oscillator strength for a multipole transition from . Equation (30), with its leading term involving the dipole polarizability is not absolutely convergent in Dalgarno and Lewis (1956). At any finite , continued summation of the series given by Eq. (30), with respect to , will eventually result in a divergence in the value of the polarization potential.
where the non-adiabatic dipole polarizability, is defined
The non-adiabatic interaction is repulsive for atoms in their ground states. The polarization interaction includes further adiabatic, non-adiabatic and higher order terms that contribute at the and , but there has been no systematic study of what could be referred to as the non-adiabatic expansion of the polarization potential.
When the Rydberg electron is in a state that has negligible overlap with the core (this is best achieved with the electron in high angular momentum orbitals), then the polarization interaction usually provides the dominant contribution to this energy shift. Suppose the dominant perturbation to the long-range atomic interaction is
where and . Equation (34) omits the and terms that are included in a more complete description Drachman (1982); Drachman and Bhatia (1995); Mitroy and Safronova (2009). The energy shift due to an interaction of this type can be written
where is usually the energy difference between two Rydberg states. The expectation values and are simply the differences in the radial expectations of the two states. These are easily evaluated using the identities of Bockasten Bockasten (1974). Plotting versus yields as the intercept and as the gradient. Such a graph is sometimes called a polarization plot.
Traditional spectroscopies such as discharges or laser excitation find it difficult to excite atoms into Rydberg states with . Exciting atoms into states with is best done with resonant excitation Stark ionization spectroscopy (RESIS) Lundeen (2005). RESIS spectroscopy first excites an atomic or ionic beam into a highly-excited state, and then uses a laser to excite the system into a very highly-excited state which is Stark ionized.
While this approach to extracting polarizabilities from Rydberg series energy shifts is appealing, there are a number of perturbations that act to complicate the analysis. These include relativistic effects , Stark shifts from ambient electric fields , second-order effects due to relaxation of the Rydberg electron in the field of the polarization potential Drake and Swainson (1991); Swainson and Drake (1992); Mitroy (2008), and finally the corrections due to the and terms, , and . Therefore, the energy shift between two neighbouring Rydberg states is
One way to solve the problem is to simply subtract these terms from the observed energy shift, e.g.
and then deduce and from the polarization plot of the corrected energy levels Mitroy and Safronova (2009).
iii.10 Stark shift measurements of polarizability differences
The Stark shift experiment predates the formulation of quantum mechanics in its modern form Stark (1913). An atom is immersed in an electric field, and the shift in wavelength of one of its spectral lines is measured as a function of the field strength. Stark shift experiments effectively measure the difference between the polarizability of the two atomic states involved in the transition. Stark shifts can be measured for both static and dynamic electric fields. While there have been many Stark shift measurements, relatively few have achieved an overall precision of 1 or better.
While the polarizabilities can generally be extracted from the Stark shift measurement, it is useful to compare the experimental values directly with theoretical predictions where high precision is achieved for both theory and experiment. In this review, comparisons of the theoretical static polarizability differences for the resonance transitions involving the alkali atoms with the corresponding Stark shifts are provided in Section V. Some of the alkali atom experiments report precisions between 0.01 and 0.1 a.u. Hunter et al. (1988, 1991, 1992); Bennett et al. (1999). The many Stark shift experiments involving Rydberg atoms van Wijngaarden (1999) are not detailed here.
Selected Stark shifts for some non-alkali atoms that are of interest for applications described in this review are discussed in Section V as well. The list is restricted to low-lying excited states for which high precision Stark shifts are available. When compared with the alkali atoms, there are not that many measurements and those that have been performed have larger uncertainties.
The tensor polarizability of an open shell atom can be extracted from the difference in polarizabilities between the different magnetic sub-levels. Consequently, tensor polarizabilities do not rely on absolute polarizability measurements and can be extracted from Stark shift measurements by tuning the polarization of a probe laser. Tensor polarizabilities for a number of states of selected systems are discussed in Section V.
iii.11 AC Stark shift measurements
There are few experimental measurements of AC Stark shifts at optical frequencies. Two recent examples would be the determination of the Stark shift for the Al clock transition Rosenband et al. (2006) and the Li - Stark shifts Sánchez et al. (2009) at the frequencies of the pump and probe laser of a two-photon resonance transition between the two states. One difficulty in the interpretation of AC Stark shift experiments is the lack of precise knowledge about the overlap of the laser beam with atoms in the interaction region. This is also a complication in the analysis of experiment on deflection of atomic beams by lasers Kadar-Kallen and Bonin (1992, 1994).
Iv Practical calculation of atomic polarizabilities
There have been numerous theoretical studies of atomic and ionic polarizabilities in the last several decades. Most methods used to determine atomic wave functions and energy levels can be adapted to generate polarizabilities. These have been divided into a number of different classes that are listed below. We give a brief description of each approach. It should be noted that the list is not exhaustive, and the emphasis here has been on those methods that have achieved the highest accuracy or those methods that have been applied to a number of different atoms and ions.
iv.1 Configuration interaction
The configuration interaction (CI) method Hibbert (1975) and its variants are widely used for atomic structure calculations owing to general applicability of the CI method. The CI wave function is written as a linear combination of configuration state functions
i.e. a linear combination of Slater determinants from a model subspace Dzuba et al. (1996). Each configuration is constructed with consideration given to anti-symmetrization, angular momentum and parity requirements. There is a great deal of variety in how the CI approach is implemented. For example, sometimes the exact functional form of the orbitals in the excitation space is generated iteratively during successive diagonalization of the excitation basis. Such a scheme is called the multi-configuration Hartree-Fock (MCHF) or multi-configuration self consistent field (MCSCF) approach Froese Fischer et al. (1997). The relativistic version of MCHF is referred to as multi-configuration Dirac-Fock (MCDF) method Grant (2007).
The CI approach has a great deal of generality since there are no restrictions imposed upon the virtual orbital space and classes of excitations beyond those limited by the computer resources. The method is particularly useful for open shell systems which contain a number of strongly interacting configurations. On the other hand, there can be a good deal of variation in quality between different CI calculations for the same system, because of the flexibility of introducing additional configuration state functions.
The most straightforward way to evaluate polarizability within the framework of the CI method it to use a direct approach by solving the inhomogeneous equation (6). RPA corrections to the dipole operator can be incorporated using the effective operator technique described in Section IV.7. It is also possible to use CI-generated matrix elements and energies to evaluate sums over states. The main drawback of the CI method is its loss of accuracy for heavier systems. It becomes difficult to include a sufficient number of configurations for heavier systems to produce accurate results even with modern computer facilities. One solution of this problem is to use a semi-empirical core potential (CICP method) described in the next subsection. Another, ab initio solution, involves construction of the effective Hamiltonian using either many-body perturbation theory (CI+MBPT) or all-order linearized coupled-cluster method (CI+all-order) and carrying out CI calculations in the valence sector. These approaches are described in the last two sections of this chapter.
iv.2 CI calculations with a semi-empirical core potential (CICP)
The ab initio treatment of core-valence correlations greatly increases the complexity of any structure calculation. Consequently, to include this physics in the calculation, using a semi-empirical approach is an attractive alternative for an atom with a few valence electrons Laughlin and Victor (1988); Müller et al. (1984); Mitroy and Bromley (2003).
In this method, the active Hamiltonian for a system with two valence electrons is written as
The and represent the direct and exchange interactions with the core electrons. In some approaches, these terms are represented by model potentials, Victor et al. (1976); Norcross and Seaton (1976); Santra et al. (2004). More refined approaches evaluate and using core wave functions calculated with the Hartree-Fock (or Dirac-Fock) method Müller et al. (1984); Migdalek and Baylis (1978); Mitroy and Bromley (2003). The one-body polarization interaction is semi-empirical in nature and can be written in its most general form as an -dependent potential, e.g.
where is the static dipole polarizability of the core and is a cutoff function that eliminates the singularity at the origin. The cutoff functions usually include an adjustable parameter that is tuned to reproduce the binding energies of the valence states. The two-electron or di-electronic polarization potential is written
There is variation between expressions for the core polarization potential, but what is described above is fairly representative. One choice for the cutoff function is Mitroy and Bromley (2003), but other choices exist.
A complete treatment of the core-polarization corrections also implies that corrections have to be made to the multipole operators Hameed et al. (1968); Hameed (1972); Müller et al. (1984); Mitroy and Bromley (2003). The modified transition operator is obtained from the mapping
Usage of the modified operator is essential to the correct prediction of the oscillator strengths. For example, it reduces the K oscillator strength by 8 Müller et al. (1984).
One advantage of this configuration interaction plus core-polarization (CICP) approach is in reducing the size of the calculation. The elimination of the core from active consideration permits very accurate solutions of the Schrödinger equation for the valence electrons. Introduction of the core-polarization potentials, and , introduces an additional source of uncertainty into the calculation. However, this additional small source of uncertainty is justified by the almost complete elimination of computational uncertainty in the solution of the resulting simplified Schrödinger equation.
The CICP approach only gives the polarizability of the valence electrons. Core polarizabilities are typically quite small for the group I and II atoms, e.g. the cesium atom has a large core polarizability of about 15.6 Zhou and Norcross (1989), but this represents only 4 of the total ground state polarizability of 401 Amini and Gould (2003). Hence, usage of moderate accuracy core polarizabilities sourced from theory or experiment will lead to only small inaccuracies in the total polarizability.
Most implementation of the CICP approach to the calculation of polarizabilities have been within a non-relativistic framework. A relativistic variant (RCICP) has recently been applied to zinc, cadmium, and mercury Ye and Wang (2008). It should be noted that even non-relativistic calculations incorporate relativistic effects to some extent. Tuning the core polarization correction to reproduce the experimental binding energy partially incorporates relativistic effects on the wave function.
iv.3 Density functional theory
Approaches based on Density Functional Theory (DFT) are not expected to give polarizabilities as accurate as those coming from the refined ab initio calculations described in the following sections. Polarizabilities from DFT calculations are most likely to be useful for systems for which large scale ab initio calculations are difficult, e.g. the transition metals. DFT calculations are often much less computationally expensive than ab initio calculations. There have been two relatively extensive DFT compilations Doolen (1984); Chu and Dalgarno (2004) that have reported dipole polarizabilities for many atoms in the periodic table.
iv.4 Correlated basis functions
The accuracy of atomic structure calculations can be dramatically improved by the use of basis functions which explicitly include the electron-electron coordinate. The most accurate calculations reported for atoms and ions with two or three electrons have typically been performed with exponential basis functions including the inter-electronic coordinates as a linear factor. A typical Hylleraas basis function for lithium would be
Difficulties with performing the multi-center integrals have effectively precluded the use of such basis functions for systems with more than three electrons. Within the framework of the non-relativistic Schrödinger equation, calculations with Hylleraas basis sets achieve accuracies of 13 significant digits Pachucki and Sapirstein (2001) for polarizability of two-electron systems and 6 significant digits for the polarizability of three-electron systems Yan et al. (1996); Tang et al. (2009a). Inclusion of relativistic and quantum electrodynamic (QED) corrections to the polarizability of helium has been carried out in Refs. Pachucki and Sapirstein (2001); Łach et al. (2004), and the resulting final value is accurate to 7 significant digits.
Another correlated basis set that has recently found increasingly widespread use utilizes the explicitly correlated gaussian (ECG). A typical spherically symmetric explicitly correlated gaussian for a three-electron system is written as Suzuki and Varga (1998)
The multi-center integrals that occur in the evaluation of the Hamiltonian can be generally reduced to analytic expressions that are relatively easy to compute. Calculations using correlated gaussians do not achieve the same precision as Hylleraas forms, but are still capable of achieving much higher precision than orbital based calculations provided the parameters and are well optimized Suzuki and Varga (1998); Komasa (2002).
iv.5 Many-body perturbation theory
The application of many-body perturbation theory (MBPT) is discussed in this section in the context of the Dirac equation. While MBPT has been applied with the non-relativistic Schrödinger equation, many recent applications most relevant to this review have been using a relativistic Hamiltonian.
The point of departure for the discussions of relativistic many-body perturbation theory (RMBPT) calculations is the no-pair Hamiltonian obtained from QED by Brown and Ravenhall (1951), where the contributions from negative-energy (positron) states are projected out. The no-pair Hamiltonian can be written in second-quantized form as , where
and a c-number term that just provides an additive constant to the energy of the atom has been omitted.
In Eqs. (45 - 46), and are creation and annihilation operators for an electron state , and the summation indices range over electron bound and scattering states only. Products of operators enclosed in brackets, such as , designate normal products with respect to a closed core. The core DHF potential is designated by and its Breit counterpart is designated by . The quantity in Eq. (45) is the eigenvalue of the Dirac equation. The quantities and in Eq. (46) are two-electron Coulomb and Breit matrix elements, respectively
where are Dirac matrices.
For neutral atoms, the Breit interaction is often a small perturbation that can be ignored compared to the Coulomb interaction. In such cases, it is particularly convenient to choose the starting potential to be the core DHF potential ,
since with this choice, the second term in Eq. (46) vanishes. The index refers to all core orbitals. The Breit term is defined as
For monovalent atoms, the lowest-order wave function is written as
where is the closed core wave function, being the vacuum wave function, and being a valence-state creation operator. The indices and refer to core orbitals.
The perturbation expansion for the wave function leads immediately to a perturbation expansion for matrix elements. Thus, for the one-particle operator written in the second-quantized form as
perturbation theory leads to an order-by-order expansion for the matrix element of between states and of an atom with one valence electron:
The first-order matrix element is given by the DHF value in the present case
The second-order expression for the matrix element of a one-body operator in a Hartree-Fock potential is given by
where . The summation index ranges over states in the closed core, and the summation index ranges over the excited states. The complete third-order MBPT expression for the matrix elements of monovalent systems was given in Ref. Johnson et al. (1996). The monumental task of deriving and evaluating the complete expression for the fourth-order matrix elements has been carried out for Na in Ref. Cannon and Derevianko (2004).
The polarizabilities are obtained using a sum-over-state approach by combining the resulting matrix elements and either experimental or theoretical energies. The calculations are carried out with a finite basis set, resulting in a finite sum in the sum-over-state expression that it is equivalent to the inclusion of all bound states and the continuum. Third-order MBPT calculation of polarizabilities is described in detail, for example, in Ref. Safronova and Safronova (2009) for Yb.
The relativistic third-order many-body perturbation theory generally gives good results for electric-dipole (E1) matrix elements of lighter systems in the cases when the correlation corrections are not unusually large. For example, the third-order value of the Na matrix element agrees with high-precision experiment to 0.6% Safronova and Johnson (2008). However, the third-order values for the matrix element in Cs and matrix element in Fr differ from the experimental data by 1.3% and 2%, respectively Safronova and Johnson (2008). For some small matrix elements, for example in Cs, third-order perturbation theory gives much poorer values. As a result, various methods that are equivalent to summing dominant classes of perturbation theory terms to all orders have to be used to obtain precision values, in particular when sub-percent accuracy is required.
The relativistic all-order correlation potential method that enables efficient treatment of dominant core-valence correlations was developed in Ref. Dzuba et al. (1989). It was used to study fundamental symmetries in heavy atoms and to calculate atomic properties of alkali-metal atoms and isoelectronic ions (see, for example, Refs. Dzuba et al. (2001, 2002) and references therein). In the correlation potential method for monovalent systems, the calculations generally start from the relativistic Hartree-Fock method in the approximation. The correlations are incorporated by means of a correlation potential defined in such a way that its expectation value over a valence electron wave function is equal to the RMBPT expression for the correlation correction to the energy of the electron. Two classes of higher-order corrections are generally included in the correlation potential: the screening of the Coulomb interaction between a valence electron and a core electron by outer electrons, and hole-particle interactions. Ladder diagrams were included to all orders in Ref. Dzuba (2008). The correlation potential is used to build a new set of single-electron states for subsequent evaluation of various matrix elements using the random-phase approximation. Structural radiation and the normalization corrections to matrix elements are also incorporated. This approach was used to evaluate black-body radiation shifts in microwave frequency standards in Refs. Angstmann et al. (2006a, b) (see Section VII.3.5).
Another class of the all-order approaches based on the coupled-cluster method is discussed in the next subsection.
iv.6 Coupled-cluster methods
In the coupled-cluster method, the exact many-body wave function is represented in the form Coester and Kümmel (1960)
where is the lowest-order atomic wave function. The operator for an N-electron atom consists of “cluster” contributions from one-electron, two-electron, , N-electron excitations of the lowest-order wave function : . In the single-double approximation of the coupled-cluster (CCSD) method, only single and double excitation terms with and are retained. Coupled-cluster calculations which use a relativistic Hamiltonian are identified by a prefix of R, e.g. RCCSD.
The exponential in Eq. (56), when expanded in terms of the -body excitations , becomes
Actual implementations of the coupled-cluster approach and subsequent determination of polarizability vary significantly with the main source of variation being the inclusion of triple excitations or non-linear terms and use of different basis sets. These differences account for some discrepancies between different coupled-cluster calculations for the same system. It is common for triple excitations to be included perturbatively. In this review, all coupled-cluster calculations that include triples in some way are labelled as CCSDT (or RCCSDT, RLCCSDT) calculations with no further distinctions being made.
We can generally separate coupled-cluster calculations of polarizabilities into two groups, but note that details of calculations vary between different works. Implementations of the CCSDT method in the form typically used for the quantum chemistry calculations use gaussian type orbital basis sets. Care should be taken to explore the dependence of the final results on the choice and size of the basis set. The dependence of the dipole polarizability values on the quality of the basis set used has been discussed, for example, in Ref. Lim and Schwerdfeger (2004). In those calculations, the polarizabilities are generally calculated using the finite-field approach Lim and Schwerdfeger (2004); Lim et al. (2005); Lupinetti and Thakkar (2005). Consequently, such CC calculations are not restricted to monovalent systems, and RCC calculations of polarizabilities of divalent systems have been reported in Refs. Sadlej et al. (1991); Schäfer et al. (2007); Lim and Schwerdfeger (2004).
The second type of relativistic coupled-cluster calculations is carried out using the linearized variant of the coupled-cluster method (referred to as the relativistic all-order method in most references), which was first developed for atomic physics calculations and applied to He in Ref. Blundell et al. (1989a). The extension of this method to monovalent systems was introduced in Ref. Blundell et al. (1989b). We refer to this approach as the RLCCSD or RLCCSDT method Safronova and Johnson (2008). We note that RLCCSDT method includes only valence triples using perturbative approach. As noted above, all CC calculations that include triples in some way are labelled as CCSDT. The RLCCSDT method uses finite basis set of B-splines rather than gaussian orbitals. The B-spline basis sets are effectively complete for each partial wave, i.e. using a larger basis set will produce negligible changes in the results. The partial waves with are generally used. Third-order perturbation theory is used to account for higher partial waves where necessary. Very large basis sets are used, typically a total of 500 - 700 orbitals are included for monovalent systems. Therefore, this method avoids the basis set issues generally associated with other coupled-cluster calculations. The actual algorithm implementation is distinct from standard quantum chemistry codes as well.
In the linearized coupled-cluster approach, all non-linear terms are omitted and the wave function takes the form
The inclusion of the nonlinear terms within the framework of this method is described in Ref. Pal et al. (2007). Restricting the sum in Eq. (58) to single, double, and valence triple excitations yields the expansion for the wave function of a monovalent atom in state :
where the indices , , and range over all possible virtual states while indices and range over all occupied core states. The quantities , are single-excitation coefficients for core and valence electrons and and are double-excitation coefficients for core and valence electrons, respectively, are the triple valence excitation coefficients. For the monovalent systems, is generally taken to be the frozen-core potential, .
We refer to results obtained with this approach as RLCCSDT, indicating inclusion of single, double, and partial triple excitations. The triple excitations are generally included perturbatively. Strong cancellations between groups of smaller terms, for example non-linear terms and certain triple excitation terms have been found in Ref. Porsev and Derevianko (2006c). As a result, additional inclusion of certain classes of terms may not necessarily lead to more accurate values.
The matrix elements for any one-body operator given in second-quantized form by Eq. (52) are obtained within the framework of the linearized coupled-cluster method as
where and are given by the expansion (59). In the SD approximation, the resulting expression for the numerator of Eq. (60) consists of the sum of the DHF matrix element and 20 other terms that are linear or quadratic functions of the excitation coefficients Blundell et al. (1989b). The main advantage of this method is its general applicability to calculation of many atomic properties of ground and excited states: energies, electric and magnetic multipole matrix elements and other transition properties such as oscillator strengths and lifetimes, and hyperfine constants, dipole and quadrupole polarizabilities, parity-nonconserving matrix elements, electron electric-dipole-moment (EDM) enhancement factors, and coefficients, etc.
The all-order method yields results for the properties of alkali atoms Safronova et al. (1999) in excellent agreement with experiment. The application of this method to the calculation of alkali polarizabilities (using a sum-over-state approach) is described in detail in Refs. Safronova et al. (1999); Derevianko et al. (1999); Arora et al. (2007a); Arora et al. (2007).
In its present form described above, the RLCCSDT method is only applicable to the calculation of polarizabilities of monovalent systems. The work on combining the RLCCSDT approach with the CI method to create a method that is more general is currently in progress Safronova et al. (2009a) and is described in Section IV.8.
iv.7 Combined CI and many-body perturbation theory
Precise calculations for atoms with several valence electrons require an accurate treatment of valence-valence correlations. While finite-order MBPT is a powerful technique for atomic systems with weakly interacting configurations, its accuracy can be limited when the wave function has a number of strongly interacting configurations. One example occurs for the alkaline-earth atoms where there is strong mixing between the and configurations of S symmetry. For such systems, an approach combining both aspects has been developed by Dzuba et al. Dzuba et al. (1996) and later applied to the calculation of atomic properties of many other systems Porsev et al. (2001); Kozlov and Porsev (1999); Porsev and Derevianko (2006b); Porsev et al. (2008); Dzuba and Ginges (2006); Dzuba and Derevianko (2010). This composite approach to the calculation of atomic structure is often abbreviated as CI+MBPT (we use RCI+MBPT designations in this review to indicate that the method is relativistic).
For systems with more than one valence electron, the precision of the CI method is drastically limited by the sheer number of the configurations that should be included. As a result, the core-core and core-valence correlations might only receive a limited treatment, which can lead to a significant loss of accuracy. The RCI+MBPT approach provides a complete treatment of core correlations to a limited order of perturbation theory. The RCI+MBPT approach uses perturbation theory to construct an effective core Hamiltonian, and then a CI calculation is performed to generate the valence wave functions.
where contains the Coulomb (or Coulomb + Breit) matrix elements . In the RCI+MBPT approach, the one-body term is modified to include a correlation potential that accounts for part of the core-valence correlations, . Either the second-order expression for or all-order chains of such terms can be used (see, for example, Ref. Dzuba et al. (1996)). The two-body Coulomb interaction term in is modified by including the two-body part of core-valence interaction that represents screening of the Coulomb interaction by valence electrons; . The quantity is calculated in second-order MBPT Dzuba et al. (1996). The CI method is then used with the modified to obtain improved energies and wave functions.
The polarizabilities are determined using the direct approach (in the valence sector) by solving the inhomogeneous equation in the valence space, approximated from Eq. (6). For state with total angular momentum and projection , the corresponding equation is written as Kozlov and Porsev (1999)
The wave function is composed of parts that have angular momenta of . This then permits the scalar and tensor polarizability of the state to be determined Kozlov and Porsev (1999).
The construction of was described in the preceding paragraphs. The effective dipole operator includes random phase approximation (RPA) corrections and several smaller MBPT corrections described in Dzuba et al. (1998). Non-RPA corrections may be neglected in some cases Kozlov and Porsev (1999). There are several variants of the RCI+MBPT method that differ by the corrections included in the effective operators and , the functions used for the basis sets, and versions of the CI code. In some implementations of the RCI+MBPT, the strength of the effective Hamiltonian is rescaled to improve agreement with binding energies. However, this procedure may not necessarily improve the values of polarizabilities.
The contributions from the dominant transitions may be separated and replaced by more accurate experimental matrix elements when appropriate. Such a procedure is discussed in detail in Ref. Porsev et al. (2008). This hybrid RCI+MBPT approach Porsev and Derevianko (2006b, a); Hachisu et al. (2008) has been used to obtain present recommended values for the polarizabilities of the and states of Mg, Ca, Sr, Hg, and Yb needed to evaluate the blackbody radiation shifts of the relevant optical frequency standards.
iv.8 Combined CI and all-order method
The RCI+MBPT approach described in the previous section includes only a limited number of the core-valence excitation terms (mostly in second order) and deteriorates in accuracy for heavier, more complicated systems. The linearized coupled-cluster approach described in Section IV.6 is designed to treat core-core and core-valence correlations with high accuracy. As noted above, it is restricted in its present form to the calculation of properties of monovalent systems. Direct extension of this method to even divalent systems faces two major problems.
First, use of the Rayleigh-Schrödinger RMBPT for heavy systems with more than one valence electron leads to a non-symmetric effective Hamiltonian and to the problem of “intruder states” Nikolić and Lindroth (2004). Second, the complexity of the all-order formalism for matrix elements increases rapidly with the number of valence electrons. The direct extensions of the all-order approach to more complicated systems is impractical. For example, the expression for all-order matrix elements in divalent systems contains several hundred terms instead of the twenty terms in the corresponding monovalent expression. However, combining the linearized coupled-cluster approach (also referred to as the all-order method) with CI method eliminates many of these difficulties. This method (referred to as CI+all-order) was developed in Ref. Safronova et al. (2009a) and tested on the calculation of energy levels of Mg, Ca, Sr, Zn, Cd, Ba, and Hg. The prefix R is used to indicate the use of the relativistic Hamiltonian.
In the RCI+all-order approach, the effective Hamiltonian is constructed using fully converged all-order excitations coefficients , ,