Hydrogen-like atom: Difference between revisions
imported>Paul Wormer |
imported>Mark Widmer mNo edit summary |
||
| (34 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
In [[physics]] and [[chemistry]], a '''hydrogen-like atom''' (or ''hydrogenic'' atom) is an [[atom]] with one [[electron]]. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge ''e(Z-1)'', where ''Z'' is the [[atomic number]] of the atom and ''e'' is the [[elementary charge]]. A better—but never used—name would therefore be hydrogen-like [[cation]]s. | |||
Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their non-relativistic [[Schrödinger equation]] can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like [[electron orbital#atomic orbital|atomic orbitals]]. The orbitals of the different hydrogen-like atoms differ from one another in one respect only: they depend on the nuclear charge ''eZ'' (which appears in their radial part). | |||
Hydrogen-like atoms ''per se'' do not play an important role in chemistry or physics. The interest in these atoms is mainly because their Schrödinger equation can be solved analytically, in exactly the same way as the Schrödinger equation of the hydrogen atom. | |||
==Quantum numbers of hydrogen-like wave functions== | |||
The non-relativistic wave functions ([[electron orbital|orbital]]s) of hydrogen-like atoms are known analytically and are labeled by three exact [[quantum number]]s, conventionally designated ''n'', ''ℓ'', and ''m''. These quantum numbers play an important role in [[atomic physics]] and [[chemistry]], as they are useful labels for quantum mechanical states of more-electron atoms, too. Although the three quantum numbers are not exact for an atom with more than one electron, they are still approximately valid, see [[Electron_orbital#Atomic_orbitals|this article]]. Because the (for many-electron atoms) approximate quantum numbers ''n'', ''ℓ'', and ''m'' are the building bricks of the [[Aufbau principle]] (building-up principle)—the construction of the [[atomic electron configuration|electronic configuration of atoms]]—they are discussed at some length in this section. | |||
===Eigenfunctions of commuting operators=== | |||
Hydrogen-like atomic orbitals are eigenfunctions of a [[Hamiltonian]] ''H'' (energy operator) with eigenvalues proportional to 1/''n''², where ''n'' is a positive integer, referred to as [[principal quantum number]]. Observe the somewhat unexpected fact that these eigenvalues do depend ''solely'' on ''n''. | |||
The hydrogen orbitals are usually chosen such that they are simultaneously eigenfunctions of ''H'' and ''l''<sup>2</sup>, the square of the one-electron [[angular momentum (quantum)|angular momentum]] vector operator | |||
:<math> | |||
\mathbf{l} \equiv -i\hbar\, (\mathbf{r}\times \boldsymbol{\nabla}) \equiv (l_x,\; l_y,\; l_z), | |||
</math> | |||
where ℏ is [[Planck's constant]] divided by 2π, the symbol '''×''' stands for a [[cross product]], '''∇''' is the [[gradient]] operator, and '''r''' is the vector pointing from the nucleus to the electron. | |||
From [[quantum mechanics]] it is known that a necessary and sufficient condition for the existence of simultaneous eigenfunctions of ''H'' and ''l''<sup> 2</sup> is the commutation of the operators | |||
:<math> | |||
l^2 \equiv l^2_x+l^2_y+l^2_z \quad\hbox{and}\quad H. | |||
</math> | |||
These two operators indeed commute. (This is due to the spherical symmetry of ''H''.) The squared operator {{nowrap|''l''<sup> 2</sup>}} has eigenvalues proportional to ''ℓ(ℓ+1)'', where ''ℓ'' is a non-negative integer (the [[azimuthal quantum number]], also known as the [[angular momentum quantum number]]). | |||
Further, since ''l''<sup> 2</sup> commutes with the three angular momentum components ''l''<sub>''x''</sub>, ''l''<sub>''y''</sub>, and ''l''<sub>''z''</sub>, it is possible to require an orbital to be an eigenfunction of any of the three components. It is conventional to choose ''l''<sub>''z''</sub>, which has an eigenvalue proportional to an integer usually denoted by ''m'' (the so-called [[magnetic quantum number]]). | |||
Count the degenerate orbitals belonging to fixed ''n'', | |||
:<math> | |||
\sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell} 1 = \sum_{\ell=0}^{n-1} 2\ell+1 = n^2. | |||
</math> | |||
In other words, the degeneracy (maximum number of linearly independent eigenfunctions of same energy) of energy level ''n'' is equal to ''n''<sup>2</sup>. This is the dimension of the irreducible representations of the symmetry group of hydrogen-like atoms, which is SO(4), and not SO(3) as for many-electron atoms. | |||
===Summary of quantum numbers=== | |||
A hydrogen-like atomic orbital ''ψ''<sub>''nℓm''</sub> is uniquely identified by the values of the principal quantum number ''n'', the azimuthal quantum number ''ℓ'', and the magnetic quantum number ''m''. These three quantum numbers are [[natural number]]s, their definitions and ranges are: | |||
:<math> | :<math> | ||
\begin{align} | \begin{align} | ||
n &= | H\;\psi_{n\ell m} &= \frac{E_\mathrm{h}}{2n^2}\;\psi_{n\ell m},&\qquad n&=1,\ldots,\infty, &\qquad\qquad&\hbox{(principal quantum number)}\\ | ||
l &= | |||
m &= | l^2\;\psi_{n\ell m} &= \hbar^2 \ell(\ell+1)\;\psi_{n\ell m}, & \qquad \ell &=0,\ldots,n-1, &\qquad&\hbox{(azimuthal quantum number)} \\ | ||
l_z\;\psi_{n\ell m}&= \hbar m\;\psi_{n\ell m},&\qquad m &=-\ell,\ldots,\ell. &\qquad&\hbox{(magnetic quantum number)}\\ | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Here ''E''<sub>h</sub> is the [[atomic unit]] of energy, see below. Note that ''n'' ≥ ''ℓ+1''. | |||
===Indication of ''ℓ'' by letters=== | |||
It is very common to denote the orbitals of different angular momentum by different letters, 2''s''-, 3''p''-orbital, etc. For historical reasons ''ℓ'' = 0 orbitals are designated by ''s'' (sharp), ''ℓ'' = 1 by ''p'' (principal), ''ℓ'' = 2 by ''d'' (diffuse), and ''ℓ'' = 3 by ''f'' (fundamental) . For higher ''ℓ'' orbitals the alphabet is followed, while ''j'' orbitals are omitted. Thus we get the following association between letters and ''ℓ'' quantum numbers | |||
:<math> | |||
\begin{matrix} | |||
s & p & d & f & g & h & i & k \\ | |||
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ | |||
\end{matrix} | |||
</math> | |||
For instance, hydrogenic ''g''-orbitals (''ℓ''=4) start at principal quantum number ''n = 5'', so that we can speak of ''5g''-, ''6g''-, etc. orbitals, but a hydrogen-like ''4g''-orbital is not defined (i.e., does not appear as a solution of the hydrogen-like Schrödinger equation). | |||
===Spin=== | |||
The set of orbital quantum numbers must be augmented by the two-valued [[spin quantum number]] ''m<sub>''s''</sub>'' = ±½ in application of the [[exclusion principle]]. This principle restricts the allowed values of the four quantum numbers in [[electron configuration]]s of more-electron atoms: it is forbidden that two electrons have the same four quantum numbers. This is an important restriction in constructing atomic states by application of the Aufbau (building up) principle. | |||
== Schrödinger equation == | == Schrödinger equation == | ||
| Line 28: | Line 62: | ||
where | where | ||
* ε<sub>0</sub> is the [[ | * ε<sub>0</sub> is the [[Vacuum permittivity|permittivity of the vacuum]], | ||
* ''Z'' is the [[atomic number]] (charge of the nucleus in unit ''e''), | * ''Z'' is the [[atomic number]] (charge of the nucleus in unit ''e''; number of [[proton]]s in the nucleus), | ||
* ''e'' is the [[elementary charge]] (charge of an electron), | * ''e'' is the [[elementary charge]] (charge of an electron), | ||
* ''r'' is the distance of the electron from the nucleus. | * ''r'' is the distance of the electron from the nucleus. | ||
The Schrödinger equation is | The Schrödinger equation is the following eigenvalue equation of the Hamiltonian (the quantity in large square brackets): | ||
:<math> | :<math> | ||
\left[ -\frac{\hbar^2}{2 \mu} \nabla^2 + V(r) \right] \psi(\mathbf{r}) = E \psi(\mathbf{r}), | \left[ -\frac{\hbar^2}{2 \mu} \nabla^2 + V(r) \right] \psi(\mathbf{r}) = E \psi(\mathbf{r}), | ||
</math> | </math> | ||
where μ is the [[reduced mass]] of the system consisting of the electron and the nucleus. Because the electron mass is about 1836 times smaller than the mass of the lightest nucleus (the proton), the value of μ is very close to the mass of the electron ''m''<sub>e</sub> for all hydrogenic atoms. In the derivation below we will make the approximation μ = ''m''<sub>e</sub>. Since ''m''<sub>e</sub> will appear explicitly in the formulas it will be easy to correct for this approximation if necessary. | |||
In [[Solid harmonics#Derivation, relation to spherical harmonics|this article]] (in which ''l''<sup> 2 </sup> is defined without Planck's constant and imaginary unit ''i'') it is shown that | In [[Solid harmonics#Derivation, relation to spherical harmonics|this article]] (in which ''l''<sup> 2 </sup> is defined without Planck's constant and imaginary unit ''i'') it is shown that | ||
the operator ∇² expressed in spherical polar coordinates, can be written as | the operator ∇² expressed in [[spherical polar coordinates]], can be written as | ||
:<math> | :<math> | ||
\hbar^2 \nabla^2 = \frac{\hbar^2}{r}\frac{\partial^2}{\partial r^2} r - \frac{l^2}{r^2}. | \hbar^2 \nabla^2 = \frac{\hbar^2}{r}\frac{\partial^2}{\partial r^2} r - \frac{l^2}{r^2}. | ||
| Line 46: | Line 80: | ||
The wave function is written as a product of functions in the spirit of the method of [[separation of variables]]: | The wave function is written as a product of functions in the spirit of the method of [[separation of variables]]: | ||
:<math>\psi(r, \theta, \phi) = R(r)\,Y_{lm}(\theta,\phi)\,</math> | :<math>\psi(r, \theta, \phi) = R(r)\,Y_{lm}(\theta,\phi)\,</math> | ||
where < | where ''Y''<sub>''lm''</sub> are [[spherical harmonics]], which are eigenfunctions of ''l''<sup> 2</sup> with eigenvalues <math>{\scriptstyle \hbar^2 l(l+1)}</math>. | ||
Substituting this product, letting ''l''<sup> 2</sup> act on ''Y''<sub>''lm''</sub>, and dividing out ''Y''<sub>''lm''</sub>, we arrive at the following one-dimensional Schrödinger equation: | |||
:<math> | :<math> | ||
- \frac{\hbar^2}{2\mu}\left[ \frac{1}{r} \frac{d^2}{d r^2} r R(r) - \frac{l(l+1)R(r)}{r^2}\right] + V(r)R(r) = E R(r) | - \frac{\hbar^2}{2\mu}\left[ \frac{1}{r} \frac{d^2}{d r^2} r R(r) - \frac{l(l+1)R(r)}{r^2}\right] + V(r)R(r) = E R(r). | ||
</math> | </math> | ||
| Line 55: | Line 89: | ||
In addition to ''l'' and ''m'', there arises a third integer ''n'' > 0 from the boundary conditions imposed on ''R(''r'')''. The expression for the normalized wave function is: | In addition to ''l'' and ''m'', there arises a third integer ''n'' > 0 from the boundary conditions imposed on ''R(''r'')''. The expression for the normalized wave function is: | ||
:<math>\psi_{nlm} = R_{nl}(r)\, Y_{lm}(\theta,\phi) | :<math>\psi_{nlm} = R_{nl}(r)\, Y_{lm}(\theta,\phi), </math> | ||
Below it will be derived that | where ''Y''<sub>''lm''</sub>(θ,φ) is a [[spherical harmonic]]. | ||
Below it will be derived that the radial function (normalized to unity) is, | |||
:<math> R_{nl} (r) = | :<math> | ||
R_{nl} (r) = \left( \frac{2 Z}{n a_{\mu}} \right)^{3/2} | |||
\left[\frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1/2}\; e^{- Z r / {n a_{\mu}}}\; \left( \frac{2 Z r}{n a_{\mu}} \right )^{l}\; L_{n-l-1}^{2l+1}\left(\tfrac{2 Z r}{n a_{\mu}} \right). | |||
</math> | |||
Here: | |||
* <math>L_{n-l-1}^{2l+1}</math> are the [[Laguerre polynomials|generalized Laguerre polynomials]] in the definition given [[Laguerre polynomials|here]]. | * <math>L_{n-l-1}^{2l+1}</math> are the [[Laguerre polynomials|generalized Laguerre polynomials]] in the definition given [[Laguerre polynomials|here]]. | ||
* <math>a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}</math> | * <math>a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}</math> | ||
Note that ''a''<sub>μ</sub> is approximately equal to ''a''<sub>0</sub> (the [[Bohr radius]]). If the mass of the nucleus is infinite then μ = ''m''<sub>''e''</sub> and ''a''<sub>μ</sub> = ''a''<sub>0</sub>. | |||
The energy eigenvalue associated with ψ<sub>''nlm''</sub> is: | |||
:<math> E_{n} = -\frac{\mu}{2} \left( \frac{Ze^2}{4 \pi \varepsilon_0 \hbar n}\right)^2 </math>. | |||
As we pointed out above it depends only on ''n'', not on ''l'' or ''m''. | |||
===Derivation of radial function=== | ===Derivation of radial function=== | ||
As | As is shown above, we must solve the one-dimensional eigenvalue equation, | ||
:<math> | :<math> | ||
-{\hbar^2 \over 2m_e | \left[ - {\hbar^2 \over 2m_e r} {d^2\over dr^2}r +{\hbar^2 l(l+1)\over 2m_e r^2}+V(r) \right] R(r)=ER(r), | ||
</math> | </math> | ||
the radial equation becomes | where we approximated μ by ''m''<sub>e</sub>. | ||
If the substitution ''u(r)'' = ''rR(r)'' is made, the radial equation becomes | |||
:<math>-{\hbar^2 \over 2m_e} {d^2 u(r) \over dr^2} + V_{\mathrm{eff}}(r) u(r) = E u(r)</math> | :<math>-{\hbar^2 \over 2m_e} {d^2 u(r) \over dr^2} + V_{\mathrm{eff}}(r) u(r) = E u(r)</math> | ||
which is a Schrödinger equation for the function ''u(r)'' with an effective potential given by | which is a Schrödinger equation for the function ''u(r)'' with an effective potential given by | ||
:<math>V_{\mathrm{eff}}(r) = V(r) + {\hbar^2l(l+1) \over 2m_e r^2} | :<math>V_{\mathrm{eff}}(r) = V(r) + {\hbar^2l(l+1) \over 2m_e r^2}.</math> | ||
The correction to the potential ''V(r)'' is called the ''centrifugal barrier term''. | |||
In order to simplify the Schrödinger equation, we introduce the following constants that define the [[atomic unit]] of energy and length, respectively, | In order to simplify the Schrödinger equation, we introduce the following constants that define the [[atomic unit]] of energy and length, respectively, | ||
:<math> E_\ | :<math> E_\mathrm{h} = m_e \left( \frac{e^2}{4 \pi \varepsilon_0 \hbar}\right)^2 \quad\hbox{and}\quad | ||
a_{0} = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}</math>. | a_{0} = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}</math>. | ||
Substitute <math> y = Zr/a_0\ | Substitute <font style="vertical-align: 6%"><math>y = Zr/a_0\;</math></font> and <font style="vertical-align: 8%"><math>\;W = E/(Z^2 E_\mathrm{h})</math></font> into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden, | ||
:<math> \left[ -\frac{1}{2} \frac{d^2}{dy^2} + \frac{1}{2} \frac{l(l+1)}{y^2} - \frac{1}{y}\right] u_l = W u_l . | :<math> \left[ -\frac{1}{2} \frac{d^2}{dy^2} + \frac{1}{2} \frac{l(l+1)}{y^2} - \frac{1}{y}\right] u_l = W u_l . | ||
</math> | </math> | ||
Two classes of solutions of this equation exist: (i) ''W'' is negative, the corresponding eigenfunctions are square integrable and the values of ''W'' are quantized (discrete spectrum). | Two classes of solutions of this equation exist: <br> | ||
(ii) ''W'' is non-negative. Every real non-negative value of ''W'' is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as [[bound state]]s, in contrast to the class (ii) solutions that are known as ''scattering states''. | (i) ''W'' is negative, the corresponding eigenfunctions are square-integrable and the values of ''W'' are quantized (discrete spectrum).<br> | ||
(ii) ''W'' is non-negative. Every real non-negative value of ''W'' is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. <br> | |||
In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as [[bound state]]s, in contrast to the class (ii) solutions that are known as ''scattering states''. | |||
For negative ''W'' the quantity <math>\alpha \equiv 2\sqrt{-2W}</math> is real and positive. The scaling of ''y'', i.e., substitution of | For negative ''W'' the quantity <font style="vertical-align: 10%"><math>\;\alpha \equiv 2\sqrt{-2W}</math></font> is real and positive. The scaling of ''y'', i.e., substitution of <font style="vertical-align: -5%"> <math>\;x \equiv \alpha y </math></font> gives the Schrödinger equation: | ||
:<math> | :<math> | ||
\left[ \frac{d^2}{dx^2} -\frac{l(l+1)}{x^2}+\frac{2}{\alpha x} - \frac{1}{4} \right] u_l = 0, | \left[ \frac{d^2}{dx^2} -\frac{l(l+1)}{x^2}+\frac{2}{\alpha x} - \frac{1}{4} \right] u_l = 0, | ||
\quad \hbox{with}\quad x \ge 0. | \quad \hbox{with}\quad x \ge 0. | ||
</math> | </math> | ||
For < | For ''x'' → <font style="vertical-align:-20%; font-size: 180%"> ∞</font>, the inverse powers of ''x'' are negligible and a solution for large ''x'' is exp(−''x''/2). The other solution, exp(''x''/2), is physically non-acceptable. For ''x'' → 0, the inverse square power dominates and a solution for small ''x'' is ''x''<sup>''l''+1</sup>. The other solution, ''x''<sup>−''l''</sup>, is physically non-acceptable. | ||
Hence, to obtain a full range solution we substitute | Hence, to obtain a full range solution we substitute | ||
:<math> | :<math> | ||
| Line 111: | Line 149: | ||
\left[ x\frac{d^2}{dx^2} + (2l+2-x) \frac{d}{dx} +(\nu -l-1)\right] f_l(x) = 0 \quad\hbox{with}\quad \nu = (-2W)^{-\frac{1}{2}}. | \left[ x\frac{d^2}{dx^2} + (2l+2-x) \frac{d}{dx} +(\nu -l-1)\right] f_l(x) = 0 \quad\hbox{with}\quad \nu = (-2W)^{-\frac{1}{2}}. | ||
</math> | </math> | ||
Provided | Provided ν−''l''−1 is a non-negative integer, say ''k'', this equation has well-behaved (regular at the origin, vanishing for infinity) polynomial solutions written as | ||
:<math> | :<math> | ||
L^{(2l+1)}_{k}(x),\qquad k=0,1,\ldots , | L^{(2l+1)}_{k}(x),\qquad k=0,1,\ldots , | ||
| Line 117: | Line 155: | ||
which are [[Laguerre polynomials#Generalized Laguerre polynomials| generalized Laguerre polynomials]] of order ''k''. We will take the convention for generalized Laguerre polynomials | which are [[Laguerre polynomials#Generalized Laguerre polynomials| generalized Laguerre polynomials]] of order ''k''. We will take the convention for generalized Laguerre polynomials | ||
of Abramowitz and Stegun.<ref name = "Abramowitz">Milton Abramowitz and Irene A. Stegun, eds. (1965). ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.'' New York: Dover. ISBN 0-486-61272-4. </ref> | of Abramowitz and Stegun.<ref name = "Abramowitz">Milton Abramowitz and Irene A. Stegun, eds. (1965). ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.'' New York: Dover. ISBN 0-486-61272-4. </ref> | ||
Note that the Laguerre polynomials given | Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah,<ref name="Messiah" >A. Messiah, ''Quantum Mechanics'', vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer</ref> | ||
are those of Abramowitz and Stegun multiplied by a factor (''2l+1+k'')! The definition given [[Laguerre polynomials#Generalized Laguerre polynomials|in this | are those of Abramowitz and Stegun multiplied by a factor (''2l+1+k'')! The definition given [[Laguerre polynomials#Generalized Laguerre polynomials|in this article]] coincides with the one of Abramowitz and Stegun. | ||
The energy becomes | The energy becomes | ||
| Line 129: | Line 167: | ||
with normalization constant | with normalization constant | ||
:<math>N_{nl} = \left[\left(\frac{2Z}{na_0}\right)^3 \cdot \frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1 \over 2}</math> | :<math>N_{nl} = \left[\left(\frac{2Z}{na_0}\right)^3 \cdot \frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1 \over 2},</math> | ||
and energy | |||
:<math> E_n = - \frac{Z^2}{2n^2}E_\ | :<math> E_n = - \frac{Z^2}{2n^2}E_\mathrm{h},\qquad n=1,2,\ldots . | ||
</math> | </math> | ||
In the computation of the normalization constant use was made of the integral | In the computation of the normalization constant use was made of the integral | ||
<ref> H. Margenau and G. M. Murphy, ''The Mathematics of Physics and Chemistry'', Van Nostrand, 2nd edition (1956), p. 130. Note that convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition | <ref> H. Margenau and G. M. Murphy, ''The Mathematics of Physics and Chemistry'', Van Nostrand, 2nd edition (1956), p. 130. Note that the convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have <math>\bar{L}^{(k)}_{n+k} = (-1)^k (n+k)! L^{(k)}_n</math>. | ||
</ref> | </ref> | ||
:<math> | :<math> | ||
\int_0^\infty x^{2l+2} e^{-x} \left[ L^{(2l+1)}_{n-l-1}(x)\right]^2 dx = | \int_0^\infty x^{2l+2} e^{-x} \left[ L^{(2l+1)}_{n-l-1}(x)\right]^2 dx = | ||
\frac{2n (n+l)!}{(n-l-1)!} . | \frac{2n (n+l)!}{(n-l-1)!} . | ||
</math> | |||
===Caveat on completeness of hydrogen-like orbitals=== | |||
In quantum chemical calculations, hydrogen-like atomic orbitals cannot serve as an expansion basis because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.<ref>This was observed as early as 1929 by E. A. Hylleraas, Z. f. Physik vol. '''48''', p. 469 (1929). English translation in H. Hettema, ''Quantum Chemistry, Classic Scientific Papers'', p. 81, World Scientific, Singapore (2000). Later it was pointed out again by H. Shull and P.-O. Löwdin, J. Chem. Phys. vol. '''23''', p. 1362 (1955).</ref> | |||
===List of radial functions=== | |||
The following list of radial functions <math>{\scriptstyle R_{nl}(r)}</math> is copied from Ref.<ref>L. Pauling and E. B. Wilson, ''Introduction to Quantum Mechanics'', McGraw-Hill, New York (1935).</ref> The scaled distance is <math>\rho_n \equiv \frac{2 Z r}{a_0 n}.</math> | |||
:<math> | |||
\begin{align} | |||
R_{10}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\; 2 \; \\ | |||
R_{20}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{2\sqrt{2} }\; \left(2-\rho_{n} \right) \\ | |||
R_{21}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{2\sqrt{6} }\; \rho_{n} \\ | |||
R_{30}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{3} }\; \left(6-6\rho_{n}+\rho_{n}^2 \right) \\ | |||
R_{31}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{6} }\; \left(4-\rho_{n} \right)\rho_{n} \\ | |||
R_{32}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{30}}\; \rho_{n}^2 \\ | |||
R_{40}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{ 96 }\; \left(24-36\rho_{n}+12\rho_{n}^2-\rho_{n}^3 \right) \\ | |||
R_{41}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{32 \sqrt{15}}\; \left(20-10\rho_{n}+\rho_{n}^2 \right)\rho_{n} \\ | |||
R_{42}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{96 \sqrt{5} }\; \left(6-\rho_{n} \right)\rho_{n}^2 \\ | |||
R_{43}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{96 \sqrt{35}}\; \rho_{n}^3 \\ | |||
R_{50}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{300\sqrt{5} }\; \left(120-240\rho_{n}+120\rho_{n}^2-20\rho_{n}^3+\rho_{n}^4 \right) \\ | |||
R_{51}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{150\sqrt{30}}\; \left(120-90\rho_{n}+18\rho_{n}^2-\rho_{n}^3 \right)\rho_{n} \\ | |||
R_{52}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{150\sqrt{70}}\; \left(42-14\rho_{n}+\rho_{n}^2 \right)\rho_{n}^2 \\ | |||
R_{53}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{300\sqrt{70}}\; \left(8-\rho_{n} \right)\rho_{n}^3 \\ | |||
R_{54}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{900\sqrt{70}}\; \rho_{n}^4 \\ | |||
\end{align} | |||
</math> | </math> | ||
==References== | ==References== | ||
<references /> | <references /> | ||
Latest revision as of 17:31, 3 November 2021
In physics and chemistry, a hydrogen-like atom (or hydrogenic atom) is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge e(Z-1), where Z is the atomic number of the atom and e is the elementary charge. A better—but never used—name would therefore be hydrogen-like cations.
Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their non-relativistic Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals. The orbitals of the different hydrogen-like atoms differ from one another in one respect only: they depend on the nuclear charge eZ (which appears in their radial part).
Hydrogen-like atoms per se do not play an important role in chemistry or physics. The interest in these atoms is mainly because their Schrödinger equation can be solved analytically, in exactly the same way as the Schrödinger equation of the hydrogen atom.
Quantum numbers of hydrogen-like wave functions
The non-relativistic wave functions (orbitals) of hydrogen-like atoms are known analytically and are labeled by three exact quantum numbers, conventionally designated n, ℓ, and m. These quantum numbers play an important role in atomic physics and chemistry, as they are useful labels for quantum mechanical states of more-electron atoms, too. Although the three quantum numbers are not exact for an atom with more than one electron, they are still approximately valid, see this article. Because the (for many-electron atoms) approximate quantum numbers n, ℓ, and m are the building bricks of the Aufbau principle (building-up principle)—the construction of the electronic configuration of atoms—they are discussed at some length in this section.
Eigenfunctions of commuting operators
Hydrogen-like atomic orbitals are eigenfunctions of a Hamiltonian H (energy operator) with eigenvalues proportional to 1/n², where n is a positive integer, referred to as principal quantum number. Observe the somewhat unexpected fact that these eigenvalues do depend solely on n.
The hydrogen orbitals are usually chosen such that they are simultaneously eigenfunctions of H and l2, the square of the one-electron angular momentum vector operator
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{l} \equiv -i\hbar\, (\mathbf{r}\times \boldsymbol{\nabla}) \equiv (l_x,\; l_y,\; l_z), }
where ℏ is Planck's constant divided by 2π, the symbol × stands for a cross product, ∇ is the gradient operator, and r is the vector pointing from the nucleus to the electron.
From quantum mechanics it is known that a necessary and sufficient condition for the existence of simultaneous eigenfunctions of H and l 2 is the commutation of the operators
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l^2 \equiv l^2_x+l^2_y+l^2_z \quad\hbox{and}\quad H. }
These two operators indeed commute. (This is due to the spherical symmetry of H.) The squared operator l 2 has eigenvalues proportional to ℓ(ℓ+1), where ℓ is a non-negative integer (the azimuthal quantum number, also known as the angular momentum quantum number).
Further, since l 2 commutes with the three angular momentum components lx, ly, and lz, it is possible to require an orbital to be an eigenfunction of any of the three components. It is conventional to choose lz, which has an eigenvalue proportional to an integer usually denoted by m (the so-called magnetic quantum number).
Count the degenerate orbitals belonging to fixed n,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell} 1 = \sum_{\ell=0}^{n-1} 2\ell+1 = n^2. }
In other words, the degeneracy (maximum number of linearly independent eigenfunctions of same energy) of energy level n is equal to n2. This is the dimension of the irreducible representations of the symmetry group of hydrogen-like atoms, which is SO(4), and not SO(3) as for many-electron atoms.
Summary of quantum numbers
A hydrogen-like atomic orbital ψnℓm is uniquely identified by the values of the principal quantum number n, the azimuthal quantum number ℓ, and the magnetic quantum number m. These three quantum numbers are natural numbers, their definitions and ranges are:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} H\;\psi_{n\ell m} &= \frac{E_\mathrm{h}}{2n^2}\;\psi_{n\ell m},&\qquad n&=1,\ldots,\infty, &\qquad\qquad&\hbox{(principal quantum number)}\\ l^2\;\psi_{n\ell m} &= \hbar^2 \ell(\ell+1)\;\psi_{n\ell m}, & \qquad \ell &=0,\ldots,n-1, &\qquad&\hbox{(azimuthal quantum number)} \\ l_z\;\psi_{n\ell m}&= \hbar m\;\psi_{n\ell m},&\qquad m &=-\ell,\ldots,\ell. &\qquad&\hbox{(magnetic quantum number)}\\ \end{align} }
Here Eh is the atomic unit of energy, see below. Note that n ≥ ℓ+1.
Indication of ℓ by letters
It is very common to denote the orbitals of different angular momentum by different letters, 2s-, 3p-orbital, etc. For historical reasons ℓ = 0 orbitals are designated by s (sharp), ℓ = 1 by p (principal), ℓ = 2 by d (diffuse), and ℓ = 3 by f (fundamental) . For higher ℓ orbitals the alphabet is followed, while j orbitals are omitted. Thus we get the following association between letters and ℓ quantum numbers
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} s & p & d & f & g & h & i & k \\ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \end{matrix} }
For instance, hydrogenic g-orbitals (ℓ=4) start at principal quantum number n = 5, so that we can speak of 5g-, 6g-, etc. orbitals, but a hydrogen-like 4g-orbital is not defined (i.e., does not appear as a solution of the hydrogen-like Schrödinger equation).
Spin
The set of orbital quantum numbers must be augmented by the two-valued spin quantum number ms = ±½ in application of the exclusion principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms: it is forbidden that two electrons have the same four quantum numbers. This is an important restriction in constructing atomic states by application of the Aufbau (building up) principle.
Schrödinger equation
The atomic orbitals of hydrogen-like atoms are solutions of the time-independent Schrödinger equation in a potential given by Coulomb's law:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = -\frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r}}
where
- ε0 is the permittivity of the vacuum,
- Z is the atomic number (charge of the nucleus in unit e; number of protons in the nucleus),
- e is the elementary charge (charge of an electron),
- r is the distance of the electron from the nucleus.
The Schrödinger equation is the following eigenvalue equation of the Hamiltonian (the quantity in large square brackets):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2 \mu} \nabla^2 + V(r) \right] \psi(\mathbf{r}) = E \psi(\mathbf{r}), }
where μ is the reduced mass of the system consisting of the electron and the nucleus. Because the electron mass is about 1836 times smaller than the mass of the lightest nucleus (the proton), the value of μ is very close to the mass of the electron me for all hydrogenic atoms. In the derivation below we will make the approximation μ = me. Since me will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.
In this article (in which l 2 is defined without Planck's constant and imaginary unit i) it is shown that the operator ∇² expressed in spherical polar coordinates, can be written as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar^2 \nabla^2 = \frac{\hbar^2}{r}\frac{\partial^2}{\partial r^2} r - \frac{l^2}{r^2}. }
The wave function is written as a product of functions in the spirit of the method of separation of variables:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(r, \theta, \phi) = R(r)\,Y_{lm}(\theta,\phi)\,}
where Ylm are spherical harmonics, which are eigenfunctions of l 2 with eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\scriptstyle \hbar^2 l(l+1)}} . Substituting this product, letting l 2 act on Ylm, and dividing out Ylm, we arrive at the following one-dimensional Schrödinger equation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \frac{\hbar^2}{2\mu}\left[ \frac{1}{r} \frac{d^2}{d r^2} r R(r) - \frac{l(l+1)R(r)}{r^2}\right] + V(r)R(r) = E R(r). }
Wave function and energy
In addition to l and m, there arises a third integer n > 0 from the boundary conditions imposed on R(r). The expression for the normalized wave function is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{nlm} = R_{nl}(r)\, Y_{lm}(\theta,\phi), }
where Ylm(θ,φ) is a spherical harmonic. Below it will be derived that the radial function (normalized to unity) is,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{nl} (r) = \left( \frac{2 Z}{n a_{\mu}} \right)^{3/2} \left[\frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1/2}\; e^{- Z r / {n a_{\mu}}}\; \left( \frac{2 Z r}{n a_{\mu}} \right )^{l}\; L_{n-l-1}^{2l+1}\left(\tfrac{2 Z r}{n a_{\mu}} \right). }
Here:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{n-l-1}^{2l+1}} are the generalized Laguerre polynomials in the definition given here.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}}
Note that aμ is approximately equal to a0 (the Bohr radius). If the mass of the nucleus is infinite then μ = me and aμ = a0.
The energy eigenvalue associated with ψnlm is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{n} = -\frac{\mu}{2} \left( \frac{Ze^2}{4 \pi \varepsilon_0 \hbar n}\right)^2 } .
As we pointed out above it depends only on n, not on l or m.
Derivation of radial function
As is shown above, we must solve the one-dimensional eigenvalue equation,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ - {\hbar^2 \over 2m_e r} {d^2\over dr^2}r +{\hbar^2 l(l+1)\over 2m_e r^2}+V(r) \right] R(r)=ER(r), }
where we approximated μ by me. If the substitution u(r) = rR(r) is made, the radial equation becomes
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -{\hbar^2 \over 2m_e} {d^2 u(r) \over dr^2} + V_{\mathrm{eff}}(r) u(r) = E u(r)}
which is a Schrödinger equation for the function u(r) with an effective potential given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{\mathrm{eff}}(r) = V(r) + {\hbar^2l(l+1) \over 2m_e r^2}.}
The correction to the potential V(r) is called the centrifugal barrier term.
In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\mathrm{h} = m_e \left( \frac{e^2}{4 \pi \varepsilon_0 \hbar}\right)^2 \quad\hbox{and}\quad a_{0} = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}} .
Substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = Zr/a_0\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \;W = E/(Z^2 E_\mathrm{h})} into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{1}{2} \frac{d^2}{dy^2} + \frac{1}{2} \frac{l(l+1)}{y^2} - \frac{1}{y}\right] u_l = W u_l . }
Two classes of solutions of this equation exist:
(i) W is negative, the corresponding eigenfunctions are square-integrable and the values of W are quantized (discrete spectrum).
(ii) W is non-negative. Every real non-negative value of W is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable.
In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as scattering states.
For negative W the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \;\alpha \equiv 2\sqrt{-2W}} is real and positive. The scaling of y, i.e., substitution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \;x \equiv \alpha y } gives the Schrödinger equation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \frac{d^2}{dx^2} -\frac{l(l+1)}{x^2}+\frac{2}{\alpha x} - \frac{1}{4} \right] u_l = 0, \quad \hbox{with}\quad x \ge 0. }
For x → ∞, the inverse powers of x are negligible and a solution for large x is exp(−x/2). The other solution, exp(x/2), is physically non-acceptable. For x → 0, the inverse square power dominates and a solution for small x is xl+1. The other solution, x−l, is physically non-acceptable. Hence, to obtain a full range solution we substitute
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(x) = x^{l+1} e^{-x/2}f_l(x).\, }
The equation for fl(x) becomes,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ x\frac{d^2}{dx^2} + (2l+2-x) \frac{d}{dx} +(\nu -l-1)\right] f_l(x) = 0 \quad\hbox{with}\quad \nu = (-2W)^{-\frac{1}{2}}. }
Provided ν−l−1 is a non-negative integer, say k, this equation has well-behaved (regular at the origin, vanishing for infinity) polynomial solutions written as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^{(2l+1)}_{k}(x),\qquad k=0,1,\ldots , }
which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials of Abramowitz and Stegun.[1] Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah,[2] are those of Abramowitz and Stegun multiplied by a factor (2l+1+k)! The definition given in this article coincides with the one of Abramowitz and Stegun.
The energy becomes
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = -\frac{1}{2n^2}\quad \hbox{with}\quad n \equiv k+l+1 . }
The principal quantum number n satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge l+1} , or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l \le n-1} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 2/n} , the total radial wavefunction is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{nl}(r) = N_{nl} \left(\tfrac{2Zr}{na_0}\right)^{l}\; e^{-{\textstyle \frac{Zr}{na_0}}}\; L^{(2l+1)}_{n-l-1}\left(\tfrac{2Zr}{na_0}\right), }
with normalization constant
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{nl} = \left[\left(\frac{2Z}{na_0}\right)^3 \cdot \frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1 \over 2},}
and energy
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n = - \frac{Z^2}{2n^2}E_\mathrm{h},\qquad n=1,2,\ldots . }
In the computation of the normalization constant use was made of the integral [3]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\infty x^{2l+2} e^{-x} \left[ L^{(2l+1)}_{n-l-1}(x)\right]^2 dx = \frac{2n (n+l)!}{(n-l-1)!} . }
Caveat on completeness of hydrogen-like orbitals
In quantum chemical calculations, hydrogen-like atomic orbitals cannot serve as an expansion basis because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.[4]
List of radial functions
The following list of radial functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\scriptstyle R_{nl}(r)}} is copied from Ref.[5] The scaled distance is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_n \equiv \frac{2 Z r}{a_0 n}.}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} R_{10}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\; 2 \; \\ R_{20}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{2\sqrt{2} }\; \left(2-\rho_{n} \right) \\ R_{21}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{2\sqrt{6} }\; \rho_{n} \\ R_{30}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{3} }\; \left(6-6\rho_{n}+\rho_{n}^2 \right) \\ R_{31}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{6} }\; \left(4-\rho_{n} \right)\rho_{n} \\ R_{32}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{9 \sqrt{30}}\; \rho_{n}^2 \\ R_{40}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{ 96 }\; \left(24-36\rho_{n}+12\rho_{n}^2-\rho_{n}^3 \right) \\ R_{41}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{32 \sqrt{15}}\; \left(20-10\rho_{n}+\rho_{n}^2 \right)\rho_{n} \\ R_{42}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{96 \sqrt{5} }\; \left(6-\rho_{n} \right)\rho_{n}^2 \\ R_{43}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{96 \sqrt{35}}\; \rho_{n}^3 \\ R_{50}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{300\sqrt{5} }\; \left(120-240\rho_{n}+120\rho_{n}^2-20\rho_{n}^3+\rho_{n}^4 \right) \\ R_{51}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{150\sqrt{30}}\; \left(120-90\rho_{n}+18\rho_{n}^2-\rho_{n}^3 \right)\rho_{n} \\ R_{52}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{150\sqrt{70}}\; \left(42-14\rho_{n}+\rho_{n}^2 \right)\rho_{n}^2 \\ R_{53}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{300\sqrt{70}}\; \left(8-\rho_{n} \right)\rho_{n}^3 \\ R_{54}(r) &= \left(\frac{Z}{a_0}\right)^{3/2} e^{-\rho_n/2}\;\frac{1}{900\sqrt{70}}\; \rho_{n}^4 \\ \end{align} }
References
- ↑ Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.
- ↑ A. Messiah, Quantum Mechanics, vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer
- ↑ H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, Van Nostrand, 2nd edition (1956), p. 130. Note that the convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{L}^{(k)}_{n+k} = (-1)^k (n+k)! L^{(k)}_n} .
- ↑ This was observed as early as 1929 by E. A. Hylleraas, Z. f. Physik vol. 48, p. 469 (1929). English translation in H. Hettema, Quantum Chemistry, Classic Scientific Papers, p. 81, World Scientific, Singapore (2000). Later it was pointed out again by H. Shull and P.-O. Löwdin, J. Chem. Phys. vol. 23, p. 1362 (1955).
- ↑ L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, McGraw-Hill, New York (1935).