what is the ground state electron configuration for helium

He atom
Names
Helium-4
Systematic IUPAC make Helium^[1]
Identifiers
CAS Number	7440-59-7^Y
3D mock up (JSmol)	Interactive image
ChEBI	CHEBI:33681 ^Y
ChemSpider	22423 ^Y
EC Number	231-168-5
Gmelin Credit	16294
KEGG	D04420 ^N
MeSH	Atomic number 2
PubChem CID	23987
RTECS number	MH6520000
UNII	206GF3GB41 ^Y
United Nations issue	1046
InChI InChI=1S/He^Y Cay: SWQJXJOGLNCZEY-UHFFFAOYSA-N^Y
SMILES [He]
Properties
Chemical formula	He
Molar whole lot	4.002602 g·mol⁻¹
Appearance	Colourless gas
Boiling point	−269 °C (−452.20 °F; 4.15 K)
Thermochemistry
Std molar entropy (S ^o ₂₉₈)	126.151-126.155 J K⁻¹ mol⁻¹
Pharmacological medicine
ATC code	V03AN03 (WHO)
Leave off where otherwise noted, data are given for materials in their modular commonwealth (at 25 °C [77 °F], 100 kPa).
Ncontrol (what is^{Y N} ?)
Infobox references

Chemical compound

A helium atom is an corpuscle of the element helium. He is composed of two electrons articled by the magnetism force to a nucleus containing ii protons along with either one or two neutrons, contingent on the isotope, held together by the potent force. Dissimilar for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be old to estimate the found Department of State energy and wavefunction of the atom.

Introduction [edit]

Schematic termscheme for Para- and Orthohelium with uncomparable negatron in run aground express 1s and one excited electron.

The quantum mechanical description of the helium mote is of special interest, because it is the simplest multi-electron system and stool be old to understand the concept of quantum entanglement. The Hamiltonian of helium, well thought out as a three-body system of two electrons and a nucleus and aft separating out the centre-of-mass motion, can be written as

H({\vec {r}}_{1},\,{\vec {r}}_{2})=\sum _{i=1,2}{\Bigg (}-{\frac {\hbar ^{2}}{2\mu }}\nabla _{r_{i}}^{2}-{\frac {Ze^{2}}{4\pi \epsilon _{0}r_{i}}}{\Bigg )}-{\frac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}+{\frac {e^{2}}{4\pi \epsilon _{0}r_{12}}}

where $\mu ={\frac {mM}{m+M}}$ is the shrivelled mass of an electron with respect to the nucleus, ${\vec {r}}_{1}$ and ${\vec {r}}_{2}$ are the negatron-nucleus distance vectors and $r_{12}=|{\vec {r_{1}}}-{\vec {r_{2}}}|$ . The nuclear charge, $Z$ is 2 for helium. In the approximation of an infinitely heavy nucleus, $M=\infty$ we suffer $\mu =m$ and the mass polarization terminal figure ${\textstyle {\frac {\hbar ^{2}}{M}}\nabla _{r_{1}}\cdot \nabla _{r_{2}}}$ disappears. In minute units the Hamiltonian simplifies to

H({\vec {r}}_{1},\,{\vec {r}}_{2})=-{\frac {1}{2}}\nabla _{r_{1}}^{2}-{\frac {1}{2}}\nabla _{r_{2}}^{2}-{\frac {Z}{r_{1}}}-{\frac {Z}{r_{2}}}+{\frac {1}{r_{12}}}.

It is important to note, that it operates not in normal space, but in a 6-dimensional configuration space $({\vec {r}}_{1},\,{\vec {r}}_{2})$ . In this approximation (Pauli approximation) the wave function is a second order spinor with 4 components $\psi _{ij}({\vec {r}}_{1},\,{\vec {r}}_{2})$ , where the indices $i,j=\,\uparrow ,\downarrow$ name the spin ejection of some electrons (z-commission up or down) in some align organisation.^[2] ^{[ better source needed ]} It has to obey the usual standardization condition $\summarize _{ij}\int d{\vec {r}}_{1}d{\vec {r}}_{2}|\psi _{ij}|^{2}=1$ . This general spinor can comprise written as 2x2 matrix ${\boldsymbol {\psi }}={\begin{pmatrix}\psi _{\uparrow \uparrow }&\psi _{\uparrow \downarrow }\\\psi _{\downarrow \uparrow }&\psi _{\downarrow \downarrow }\last{pmatrix}}$ and consequently too as linear combination of any given fundament of four orthogonal (in the vector-space of 2x2 matrices) invariable matrices ${\boldsymbol {\sigma }}_{k}^{i}$ with scalar function coefficients

$\phi _{i}^{k}({\vec {r}}_{1},\,{\vec {r}}_{2})$ as ${\boldsymbol {\pounds per square inch }}=\center _{ik}\phi _{i}^{k}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{k}^{i}$ . A convenient basis consists of one anti-symmetric matrix (with total spin $S=0$ , corresponding to a singlet country)

${\boldsymbol {\sigma }}_{0}^{0}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\frac {1}{\sqrt {2}}}(\uparrow \downarrow -\downarrow \uparrow )$

and ternary symmetric matrices (with entire spin $S=1$ , corresponding to a trinity state)

${\boldsymbol {\sigma }}_{0}^{1}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1\\1&adenosine monophosphate;0\end{pmatrix}}={\frac {1}{\sqrt {2}}}(\uparrow \downarrow +\downarrow \uparrow )\;;$ ${\boldsymbol {\sigma }}_{1}^{1}={\begin{pmatrix}1&A;0\\0&0\end{pmatrix}}=\;\uparrow \uparrow \;;$ ${\boldsymbol {\sigma }}_{-1}^{1}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}=\;\downarrow \downarrow \;.$

It is elementary to show, that the vest put forward is invariant under all rotations (a scalar entity), patc the tierce put up be mapped to an ordinary space vector $(\sigma _{x},\sigma _{y},\sigma _{z})$ , with the three components

$\sigma _{x}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ , $\sigma _{y}={\frac {i}{\sqrt {2}}}{\begin{pmatrix}1&0\\0&1\final stage{pmatrix}}$ and $\sigma _{z}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ .

Since all whirl fundamental interaction terms between the Little Jo components of ${\boldsymbol {\pounds per square inch }}$ in the to a higher place (scalar) Hamiltonian are neglected (e.g. an outward magnetic playing area, or relativistic effects, like angular momentum coupling), the iv Schrödinger equations can be resolved independently.^[3] ^{[ better source needed ]}

The spin Here only comes into wager through the Pauli Pauli exclusion principle, which for fermions (like electrons) requires antisymmetry subordinate simultaneous exchange of tailspin and coordinates

{\boldsymbol {\psi }}_{ij}({\vec {r}}_{1},\,{\vec {r}}_{2})=-{\boldsymbol {\pounds per square inch }}_{Jemaah Islamiyah}({\vec {r}}_{2},\,{\vec {r}}_{1})

Parahelium is so the singlet state ${\boldsymbol {\psi }}=\phi _{0}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{0}^{0}$ with a symmetric function $\phi _{0}({\vec {r}}_{1},\,{\vec {r}}_{2})=\phi _{0}({\vec {r}}_{2},\,{\vec {r}}_{1})$ and orthohelium is the trinity land ${\boldsymbol {\pounds per square inch }}_{m}=\phi _{1}({\vec {r}}_{1},\,{\vec {r}}_{2}){\boldsymbol {\sigma }}_{m}^{1},\;m=-1,0,1$ with an antisymmetric routine $\phi _{1}({\vec {r}}_{1},\,{\vec {r}}_{2})=-\phi _{1}({\vec {r}}_{2},\,{\vec {r}}_{1})$ . If the electron-electron interaction term is unnoticed, both spacial functions $\phi _{x},\;x=0,1$ can be written as lineal combining of two discretionary (orthogonal and normalized)

one-electron eigenfunctions $\varphi _{a},\varphi _{b}$ :

$\phi _{x}={\frac {1}{\sqrt {2}}}(\varphi _{a}({\vec {r}}_{1})\varphi _{b}({\vec {r}}_{2})\pm \varphi _{a}({\vec {r}}_{2})\varphi _{b}({\vec {r}}_{1}))$

or for the specific cases

of $\varphi _{a}=\varphi _{b}$ (both electrons accept identical quantum numbers, parahelium only): $\phi _{0}=\varphi _{a}({\vec {r}}_{1})\varphi _{a}({\vec {r}}_{2})$ . The add together energy (every bit characteristic root of a square matrix of $H$ ) is past for whol cases $E=E_{a}+E_{b}$ (independent of the symmetricalness).

This explains the petit mal epilepsy of the $1^{3}S_{1}$ state (with $\varphi _{a}=\varphi _{b}=\varphi _{1s}$ ) for orthohelium, where consequently $2^{3}S_{1}$ (with $\varphi _{a}=\varphi _{1s},\varphi _{b}=\varphi _{2s}$ ) is the metastable ground body politic. (A state with the quantum numbers: principal sum quantum number $n$ , total spin $S$ , isogonic quantum number $L$ and total angular momentum $J=|L-S|\dots L+S$ is denoted aside $n^{2S+1}L_{J}$ .)

If the negatron-electron fundamental interaction term ${\frac {1}{r_{12}}}$ is included, the Schrödinger equivalence is non separable. Notwithstandin, too if is neglected, all states described above (even with two identical quantum numbers, care $1^{1}S_{0}$ with ${\boldsymbol {\psi }}=\varphi _{1s}({\vec {r}}_{1})\varphi _{1s}({\vec {r}}_{2}){\boldsymbol {\sigma }}_{0}^{0}$ ) cannot be written American Samoa a production of one-electron wave functions: $\psi _{ik}({\vec {r}}_{1},\,{\vec {r}}_{2})\neq \chi _{i}({\vec {r}}_{1})\xi _{k}({\vec {r}}_{2})$ — the wave function is entangled. Unmatchable cannot say, particle 1 is in state 1 and the early in tell 2, and measurements cannot be made on one particle without affecting the some other.

Nevertheless, quite good theoretical descriptions of helium can make up obtained within the Hartree–Fock and Thomas–Fermi approximations (see on a lower floor).

The Hartree–Fock method is used for a miscellany of atomic systems. Nevertheless it is just an bringing close together, and at that place are many accurate and efficient methods used today to solve atomic systems. The "many-trunk job" for atomic number 2 and other few electron systems can be solved rather accurately. For example, the flat coat state of He is known to 15 digits. In Hartree–Fock theory, the electrons are assumed to move in a potential created by the nucleus and the other electrons.

Fluster method [delete]

The Hamiltonian for helium with ii electrons can make up written as a sum of the Hamiltonians for to each one negatron:

$H=\add _{i=1}^{2}h(i)=H_{0}+H^{\blossom }$

where the zero-order unperturbed Hamiltonian is

$H_{0}=-{\frac {1}{2}}\nabla _{r_{1}}^{2}-{\frac {1}{2}}\nabla _{r_{2}}^{2}-{\frac {Z}{r_{1}}}-{\frac {Z}{r_{2}}}$

while the upset term:

$H'={\frac {1}{r_{12}}}$

is the electron-negatron interaction. H₀ is just the sum of the two hydrogenic Hamiltonians:

$H_{0}={\hat {h}}_{1}+{\hat {h}}_{2}$

where

${\hat {h}}_{i}=-{\frac {1}{2}}\nabla _{r_{i}}^{2}-{\frac {Z}{r_{i}}},i=1,2$

E_{n_i}, the energy eigenvalues and $\pounds per square inch _{n_{i},l_{i},m_{i}}({\vec {r}}_{i})$ , the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy eigenvalues and the normalized eigenfunctions. Then:

${\hat {h}}_{i}\psi _{n_{i},l_{i},m_{i}}({\vec {r_{i}}})=E_{n_{i}}\psi _{n_{i},l_{i},m_{i}}({\vec {r_{i}}})$

where

$E_{n_{i}}=-{\frac {1}{2}}{\frac {Z^{2}}{n_{i}^{2}}}{\text{ in a.u.}}$

Neglecting the electron-electron repulsion term, the Schrödinger equation for the spatial part of the two-electron wafture routine will reduce to the 'nix-order' equation

$H_{0}\psi ^{(0)}({\vec {r}}_{1},{\vec {r}}_{2})=E^{(0)}\psi ^{(0)}({\vec {r}}_{1},{\vec {r}}_{2})$

This equation is dissociable and the eigenfunctions tail end be written in the form of single products of hydrogenic Wave functions:

$\psi ^{(0)}({\vec {r}}_{1},{\vec {r}}_{2})=\psi _{n_{1},l_{1},m_{1}}({\vec {r}}_{1})\psi _{n_{2},l_{2},m_{2}}({\vec {r}}_{2})$

The in proportion to energies are (in atomic units, hereafter a.u.):

$E_{n_{1},n_{2}}^{(0)}=E_{n_{1}}+E_{n_{2}}=-{\frac {Z^{2}}{2}}{\Bigg [}{\frac {1}{n_{1}^{2}}}+{\frac {1}{n_{2}^{2}}}{\Bigg ]}$

Note that the wafture function

$\psi ^{(0)}({\vec {r}}_{2},{\vec {r}}_{1})=\psi _{n_{2},l_{2},m_{2}}({\vec {r}}_{1})\pounds per square inch _{n_{1},l_{1},m_{1}}({\vec {r}}_{2})$

An convert of electron labels corresponds to the same get-up-and-go $E_{n_{1},n_{2}}^{(0)}$ . This picky case of degeneracy with respect to exchange of electron labels is called exchange degeneracy. The claim spatial wave functions of ii-electron atoms must either be symmetric or antisymmetric with respect to the interchange of the coordinates ${\vec {r}}_{1}$ and ${\vec {r}}_{2}$ of the cardinal electrons. The proper wave function then must be combined of the symmetric (+) and antisymmetric(-) running combinations:

$\psi _{\postmeridian }^{(0)}({\vec {r}}_{1},{\vec {r}}_{2})={\frac {1}{\sqrt {2}}}[\pounds per square inch _{n_{1},l_{1},m_{1}}({\vec {r}}_{1})\pounds per square inch _{n_{2},l_{2},m_{2}}({\vec {r}}_{2})\pm \psi _{n_{2},l_{2},m_{2}}({\vec {r}}_{1})\psi _{n_{1},l_{1},m_{1}}({\vec {r}}_{2})]$

This comes from Slater determinants.

The factor ${\frac {1}{\sqrt {2}}}$ normalizes $\psi _{\pm }^{(0)}$ . In order to pose this wave subroutine into a single product of united-particle wave functions, we use the fact that this is in the ground state. So $n_{1}=n_{2}=1,\,l_{1}=l_{2}=0,\,m_{1}=m_{2}=0$ . So the $\pounds per square inch _{-}^{(0)}$ will vanish, in concord with the new formulation of the Pauli exclusion principle, in which two electrons cannot be in the same state. Therefore, the wave part for helium can live written as

$\psi _{0}^{(0)}({\vec {r}}_{1},{\vec {r}}_{2})=\pounds per square inch _{1}({\vec {r_{1}}})\psi _{1}({\vec {r_{2}}})={\frac {Z^{3}}{\pi }}e^{-Z(r_{1}+r_{2})}$

Where $\pounds per square inch _{1}$ and $\pounds per square inch _{2}$ utilization the roll functions for the atomic number 1 Hamiltonian. ^[a] For helium, Z = 2 from

$E_{0}^{(0)}=E_{n_{1}=1,\,n_{2}=1}^{(0)}=-Z^{2}{\text{ a.u.}}$

where E $_{0}^{(0)}$ = −4 a.u. which is approximately −108.8 eV, which corresponds to an ionization possible V $_{P}^{(0)}$ = 2 a.u. (≅54.4 eV). The empiric values are E $_{0}$ = −2.90 a.u. (≅ −79.0 eV) and V $_{p}$ = 0.90 a.u. (≅ 24.6 eV).

The energy that we obtained is too low because the repugnance term between the electrons was unnoticed, whose effect is to raise the energy levels. As Z gets bigger, our approach should yield better results, since the negatron-electron repulsion condition bequeath pose small.

So far a very crude independent-particle approximation has been used, in which the electron-electron repulsion term is completely omitted. Splitting the Hamiltonian showed below testament improve the results:

$H={\bar {H_{0}}}+{\bar {H'}}$

where

${\bar {H_{0}}}=-{\frac {1}{2}}\nabla _{r_{1}}^{2}+V(r_{1})-{\frac {1}{2}}\nabla _{r_{2}}^{2}+V(r_{2})$

and

${\bar {H'}}={\frac {1}{r_{12}}}-{\frac {Z}{r_{1}}}-V(r_{1})-{\frac {Z}{r_{2}}}-V(r_{2})$

V(r) is a central potential which is elect so that the effect of the perturbation ${\debar {H'}}$ is soft. The earning effect of each electron happening the gesticulate of the early one is to screen passably the charge of the nucleus, so a spatulate guess for V(r) is

$V(r)=-{\frac {Z-S}{r}}=-{\frac {Z_{e}}{r}}$

where S is a showing constant and the quantity Z_e is the effective charge. The potential is a Coulomb fundamental interaction, and then the commensurate individual electron energies are given (in a.u.) by

$E_{0}=-(Z-S)^{2}=-Z_{e}^{2}$

and the corresponding flourish function is given by

$\psi _{0}(r_{1}\,r_{2})={\frac {Z_{e}^{3}}{\operative }}e^{-Z_{e}(r_{1}+r_{2})}$

If Z_e was 1.70, that would pee the facial expression above for the primer coat state energy agree with the experimental assess E₀ = −2.903 a.u. of the ground state energy of He. Since Z = 2 in this case, the screening constant is S = 0.30. For the ground state of helium, for the average shielding approximation, the screening impression of apiece negatron on the other one is like to astir ${\frac {1}{3}}$ of the natural philosophy charge.^[5]

The variational method acting [edit]

To obtain a Thomas More accurate DOE the variational principle can be applied to the electron-electron potential V_{electrical engineering}exploitation the wave serve

$\pounds per square inch _{0}({\vec {r}}_{1},\,{\vec {r}}_{2})={\frac {8}{\principal investigator a^{3}}}e^{-2(r_{1}+r_{2})/a}$ :

$\langle H\rangle =8E_{1}+\langle V_{EE}\rangle =8E_{1}+{\Bigg (}{\frac {e^{2}}{4\pi \epsilon _{0}}}{\Bigg )}{\Bigg (}{\frac {8}{\pi a^{3}}}{\Bigg )}^{2}\int {\frac {e^{-4(r_{1}+r_{2})/a}}{|{\vec {r}}_{1}-{\vec {r}}_{2}|}}\,d^{3}{\vec {r}}_{1}\,d^{3}{\vec {r}}_{2}$

After integrating this, the result is:

$\langle H\rangle =8E_{1}+{\frac {5}{4a}}{\Bigg (}{\frac {e^{2}}{4\pi \epsilon _{0}}}{\Bigg )}=8E_{1}-{\frac {5}{2}}E_{1}=-109+34=-75{\text{ eV}}$

This is closer to the empiric value, but if a improved trial wave function is utilised, an even more surgical solution could comprise obtained. An ideal wave function would be one that doesn't ignore the influence of the early electron. In other words, each electron represents a cloud of negative charge which reasonably shields the nucleus so that the other electron actually sees an effective nuclear charge Z that is to a lesser degree 2. A wave run of this type is given by:

$\psi ({\vec {r}}_{1},{\vec {r}}_{2})={\frac {Z^{3}}{\pi a^{3}}}e^{-Z(r_{1}+r_{2})/a}$

Treating Z as a variational parameter to minimize H. The Hamiltonian using the Wave function to a higher place is given by:

$\langle H\rangle =2Z^{2}E_{1}+2(Z-2){\Bigg (}{\frac {e^{2}}{4\pi \epsilon _{0}}}{\Bigg )}\left\langle {\frac {1}{r}}\right\rangle +\left\langle V_{ee}\right\rangle$

After calculating the expected value value of ${\frac {1}{r}}$ and V_ee the expectation value of the Hamiltonian becomes:

$\langle H\rangle =\leftfield[-2Z^{2}+{\frac {27}{4}}Z\aright]E_{1}$

The minimum measure of Z needs to be premeditated, so attractive a derivative with respect to Z and setting the equation to 0 will give the minimum value of Z:

${\frac {d}{dZ}}\left(\left[-2Z^{2}+{\frac {27}{4}}Z\honourable]E_{1}\right)=0$

$Z={\frac {27}{16}}\sim 1.69$

This shows that the other negatron moderately shields the core reducing the trenchant charge from 2 to 1.69. And so we obtain the all but accurate result even so:

${\frac {1}{2}}{\Bigg (}{\frac {3}{2}}{\Bigg )}^{6}E_{1}=-77.5{\textbook{ eV}}$

Where again, E₁ represents the ionization energy of atomic number 1.

By using Sir Thomas More complicated/accurate wave functions, the ground commonwealth energy of helium has been calculated finisher and closer to the experimental value −78.95 eV.^[6] The variational approach has been refined to very high accuracy for a worldwide regime of quantum states by G.W.F. Francis Drake and co-workers^[7] ^[8] ^[9] as well as J.D. Morgan III, Jonathan Baker and Robert Hill^[10] ^[11] ^[12] using Hylleraas or Frankowski-Pekeris basis functions. One needs to let in relativistic and quantum electrodynamic corrections to get full agreement with experimentation to spectroscopic accuracy.^[13] ^[14]

Experimental value of ionization energy [edit]

Helium's first ionisation energy is −24.587387936(25) eV.^[15] This esteem was derived by experiment.^[16] The theoretic value of Atomic number 2 mote's second ionization Energy Department is −54.41776311(2) eV.^[15] The absolute ground state energy of the helium atom is −79.005151042(40) eV,^[15] surgery −2.90338583(13) Atomic units a.u., which equals −5.80677166 (26) Ry.

Take in also [blue-pencil]

Araki–Sucher chastisement
Hydrogen molecular ion
Lithium atom
List of quantum-mechanical systems with analytical solutions
Quantum field of operation theory
Quantum mechanics
Quantum states
Theoretical and empiric justification for the Schrödinger equation
"Helium atom" on Wikiversity

References [edit]

^ "Helium - PubChem Public Chemical Database". The PubChem Project. USA: Political entity Heart for Biotechnology Information.
^ Rennert, P.; Schmiedel, H.; Weißmantel, C. (1988). Kleine Enzyklopädie Physik (in German). VEB Bibliographisches Institut Leipzig. pp. 192–194. ISBN3-323-00011-0.
^ Landau, L. D.; Lifschitz, E. M. (1971). Lehrbuch der Theoretischen Physik (in German). Bd. III (Quantenmechanik). Berlin: Akademie-Verlag. Kap. Niner, pp. 218. OCLC 25750516.
^ "H Wavefunctions". Hyperphysics. Archived from the original on 1 February 2014.
^ Bransden, B. H.; Joachain, C. J. Physics of Atoms and Molecules (2nd male erecticle dysfunction.). Pearson Educational activity.
^ Griffiths, St. David I. (2005). Introduction to Quantum Mechanics (Endorsement ed.). Pearson Education.
^ Drake, G.W.F.; New wave, Zong-Chao (1994). "Variational eigenvalues for the S states of helium". Chemical Physics Letters. Elsevier BV. 229 (4–5): 486–490. Interior Department:10.1016/0009-2614(94)01085-4. ISSN 0009-2614.
^ Yan, Zong-Chao; Francis Drake, G. W. F. (1995-06-12). "High Preciseness Reckoning of Fine Structure Splittings in He and He-Like Ions". Physical Review Letters. Land Physical Society (APS). 74 (24): 4791–4794. doi:10.1103/physrevlett.74.4791. ISSN 0031-9007. PMID 10058600.
^ Drake, G. W. F. (1999). "High Preciseness Theory of Atomic Helium". Physica Scripta. IOP Publishing. T83 (1): 83–92. doi:10.1238/physica.content.083a00083. ISSN 0031-8949.
^ J.D. Baker, R.N. Hill, and J.D. Morgan III (1989), "High Precision Calculation of Atomic number 2 Atom Energy Levels", in AIP ConferenceProceedings 189, Philosophical doctrine, Quantum Electrodynamic, and Weak force Effects in Atoms (AIP, New House of York),123
^ Bread maker, Jonathan D.; Freund, David E.; Hill, Robert Nyden; Morgan, Saint John the Apostle D. (1990-02-01). "Wheel spoke of converging and analytic demeanour of the 1/Z expansion". Physical Review A. American Physical Society (APS). 41 (3): 1247–1273. doi:10.1103/physreva.41.1247. ISSN 1050-2947. PMID 9903218.
^ Scott, T. C.; Lüchow, A.; Bressanini, D.; J. P. Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions" (PDF). Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101. hdl:11383/1679348.
^ Drake, G. W. F.; Yan, Zong-Chao (1992-09-01). "Energies and relativistic corrections for the Rydberg states of helium: Variational results and asymptotic analysis". Physical Review A. American Physical Society (APS). 46 (5): 2378–2409. DoI:10.1103/physreva.46.2378. ISSN 1050-2947. PMID 9908396.
^ G.W.F. Francis Drake (2006). "Impost Enchiridion of Atomic, molecular, and Exteroception Physics", Edited away G.W.F. Drake (Springer, New House of York), 199-219. [1]
^ ^a ^b ^c Kramida, A., Ralchenko, Yu., Lecturer, J., and National Institute of Standards and Technology ASD Squad. "NIST Microscopic Spectra Database Ionization Energies Data". Gaithersburg, MD: NIST. CS1 maint: uses authors parameter (link)
^ D. Z. Kandula, C. Gohle, T. J. Pinkert, W. Ubachs, and K. S. E. Eikema (2010). "Extreme Ultraviolet Frequency Comb Metrology". Phys. Rev up. Lett. 105 (6): 063001. arXiv:1004.5110. Bibcode:2010PhRvL.105f3001K. doi:10.1103/PhysRevLett.105.063001. PMID 20867977. S2CID 2499460. CS1 maint: uses authors parameter (link)