Helium atom, polarizability

The first ab initio smdy of an interaction polarizability was that of O Brien et al. (1973) on a pair of helium atoms. They obtained /0(r) for the range r = 3.5ao through lOao- The experimentally determined value of is negative, which suggests that the incremental mean pair polarizability must be negative around the minimum in the potential curve. 

Table 5 De-excitation Cross Sections (cm) at a Mean Collisional Energy Corresponding to Room Temperature (295 K) and De-excitation Probabilities (P) of Excited Helium Atoms He (He = He(2 S), He(2 S), and He(2 P)) and polarizabilities (am) of target molecules (From Ref. 154.)...

The electron distribution of the helium atom in field-free space is, of course, spherically symmetric. The atom has, however, a large polarizability in a quadrupole electric field, which we may ascribe to the partial orientation of the prolate ellipsoid. 

DuPre and McTague have used a Hirschfelder-Iinnett wave function for the triplet state of the hydrogen molecule to approximate the wave function of a pair of helium atoms. On evaluating an approximate expres-aon for the polarizability they find Icqs < oo at short range and > o at long range. 

We have calculated the second- and fourth-order dipole susceptibilities of an excited helium atom. Numerical results have been obtained for the ls2p Pq-and ls2p f2-states of helium. For the accurate calculations of these quantities we have used the model potential method. The interaction of the helium atoms with the external electric held F is treated as a perturbation to the second- and to the fourth orders. The simple analytical expressions have been derived which can be used to estimate of the second- and higher-order matrix elements. The present set of numerical data, which is based on the Green function method, has smaller estimated uncertainties in ones than previous works. This method is developed to high-order of the perturbation theory and it is shown specihcally that the continuum contribution is surprisingly large and corresponds about 23% for the scalar part of polarizability. 

The helium pair polarizability increment has been studied extensively [19, 29, 39, 41, 48, 56, 57, 63]. We mention in particular the most recent work by Dacre [48], which includes a careful review of previous results. Systems such as H-H in the and states may be considered as tractable examples representative of various types of real collisional pairs [6,28,54, 55,66,74,85, 144, 145, 147]. Elaborate self-consistent field (SCF) calculations, supplemented by configuration interaction (Cl) corrections are also known for the neon diatom [50]. For atom pairs with more electrons, attempts have been made to correct the SCF data in some empirical fashion for Cl effects [47, 49]. Ab initio studies of molecular systems, such as H2-H2 and N2-N2 have been communicated [16, 18]. 

The data points are fitted in a least-square sense to a fourth degree polynomial, and the properties thereby obtained are presented in Table 3. Since the atom possesses spherical symmetry there is only a single independent component of the a-tensor as well as the y-tensor. The curvature of the energy, or the polarizability, at the SCF level differs by less than 5% compared to the FCI result, and the MP2 value captures slightly more than half of the correlation effect. Electron correlation plays a more important role in the determination of the fourth-order property y. Again the MP2 method captures slightly more than half of the total contribution, which amounts to 21% at the FCI level of theory. The trends we have seen here in the example of the helium atom are more or less representative for closed-shell molecules in general. 

Here we consider [25] the properties of H at the centre of a spherical box of radius R, using a numerical approach to obtain the energies and polarizabilities. We also develop some model wave functions, simple expressions for the energies and polarizability, deduce the critical radius R for which E = 0, and extend the analysis to the confined helium atom with effective screening. 

Marin and Cruz [16] also computed some properties of physical interest for the CHA, such as the Fermi contact term [30,34,35,38], diamagnetic screening constant [30,34,35], polarizability [30,34,35,44,45,57-61,63] and pressure [1,30,34,35]. In addition, they studied the hydrogen and helium atom in penetrable boxes, and the hydrogen atom between parallel hard walls [16]. 

In a second work [100], they studied the polarization of the confined helium atom via the Kirkwood approach [57], finding a decreasing tendency for polarizability as the pressure grows. 

Polarizability allows gases containing atoms or nonpolar molecules (for example, He and N2) to condense. In a helium atom the electrons are moving at some distance from the nucleus. At any instant it is likely that the atom has a dipole moment created by the specific positions of the electrons. This dipole moment is called an instantaneous dipole because it lasts for just a tiny fraction of a second. In the next instant the electrons are in different locations and the atom has a new instantaneous dipole, and so on. Averaged over time (that is, the time it takes to make a dipole moment measurement), however, the atom has no dipole moment because the instantaneous dipoles... 

A quantum mechanical interpretation of temporary dipoles was provided by Fritz London in 1930. London showed that the magnitude of this attractive interaction is directly proportional to the polarizability of the atom or molecule. As we might expect, dispersion forces may be quite weak. This is certainly true for helium, which has a boiling point of only 4.2 K, or 269°C. (Note that hehum has only two electrons, which are tightly held in the Is orbital. Therefore, the helium atom has low polarizability.)... 

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

Table 29 3.—Variation Functions for the Calculation of the Polarizability of the Normal Helium Atom Experimental value a = 0.205 10 24 cm3...

Problem 29-3. Using the method of Section 27e and the screening-constant wave function 2 of Table 29-1, evaluate the polarizability of the helium atom, taking as the zero point for energy the singly ionized atom. 

Hass6 has considered five variation functions of this form, shown with their results in Table 47-2. The success of his similar treatment of the polarizability of helium (function 6 of Table 29-3) makes it probable that the value — 1.413e2aj /i28 for W" is not in error by more than a few per cent. Slater and Kirkwood1 obtained values 1.13, 1.78, and 1.59 for the coefficient of — e2al/R6 by the use of variation functions based on their helium atom functions mentioned in Section 29e. An approximate discussion of dipole-quadrupole and quadrupole-quadrupole interactions has been given by Margenau.1... 

The next phase for the theorists in connection with this work lies in predictions of helium atom scattering intensities associated with surface phonon creation and annihilation for each variety of vibrational motion. In trying to understand why certain vibrational modes in these similar materials appear so much more prominently in some salts than others, one is always led back to the guiding principle that the vibrational motion has to perturb the surface electronic structure so that the static atom-surface potential is modulated by the vibration. Although the polarizabilities of the ions may contribute far less to the overall binding energies of alkali halide crystals than the Coulombic forces do, they seem to play a critical role in the vibrational dynamics of these materials. 

The ionic insulators discussed in some detail in the previous section have closed shell electronic configurations similar to the noble gases and electronic distributions which are localized around the electronic core. The principal interactions are Coulombic, although their polarizabilities appear to influence greatly the response of the electronic distribution to surface lattice vibrations. For other materials, particularly metals and some layered compounds, the conduction and valence electrons are best thought of as somewhat delocalized if not entirely free. These electrons are what the helium atoms scatter from, and their states of motion are significantly modulated by the vibrations of the atomic cores. Thus, for these materials HAS is very... 

The QED calculations performed to date have been focused on the energy. The first calculations of atomic susceptibilities (helium) within an accuracy including the c terms were carried out independently by Pachucki and Sapirstein and by Cencek and coworkers, and with accuracy up to c (with estimation of the c term) by Each and coworkers. To get a sense of what subtle effects may be computed nowadays. Table 3.1 shows the components of the first ionization energy and of the dipole polarizability (see Chapter 12) of the helium atom. 

Table 3.1. Contributions of various physical effects (non-relativistic, Bieit, QED, and beyond QED, distinct physical contributions shown in bold) to the ionization energy and the dipole polarizability a of the helium atom, as well as comparison with the experimental values (all quantities are expressed in atomic units i.e.. e = 1. fi = 1, mo = 1- where iiiq denotes the rest mass of the electron). The first column gives the symbol of the term in the Breit-Pauli Hamiltonian [Eq. (3.72)] as well as of the QED corrections given order by order (first corresponding to the electron-positron vacuum polarization (QED), then, beyond quantum electrodynamics, to other particle-antiparticle pairs (non-QED) li,7T,. ..) split into several separate effects. The second column contains a short description of the effect. The estimated error (third and fourth columns) is given in parentheses in the units of the last figure reported. 

What about the creation of other (than e-p) particle-antiparticle pairs from the vacuum the larger the rest mass is, the more difficult it is to squeeze out the corresponding particle-antiparticle pair. And yet we have some tiny effect (see non-QED entry) corresponding to the creation of such pairs as muon-antimuon (jx), pion-antipion (tt), etc. This means that the helium atom is composed of the nucleus and the two electrons only, when we look at it within a certain approximation. To tell the truth, the atom contains also photons, electrons, positrons, muons, pions, and whatever you wish, but with a smaller and smaller probability of appearance. All that has only a minor effect of the order of something like the seventh significant figure (both for the ionization potential and for the polarizability). 

P.W. Fowler, K.L.C. Hunt, H.M. Kelly, A.J. Sadley, Multipole polarizabilities of the helium atom and collision-induced polarizabilities of pairs containing He or H atoms. J. Chem. Phys. 100(4), 2932-2935 (1994)... 

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