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Helium atom

This article is about the physics of atomic helium. For other properties of helium, seehelium.

Helium is an element and the next simplest atom to solve after the hydrogen atom. Helium is composed of two electrons in orbit around a nucleus containing two protons along with either one or two neutrons, depending on the isotope. The hydrogen atom is used extensively to aid in solving the helium atom. The Niels Bohr model of the atom gave a very accurate explanation of the hydrogen spectrum, but when it came to helium it collapsed. Werner Heisenberg developed a modification of Bohr's analysis but it involved half-integral values for the quantum numbers. Thomas-Fermi theory, also known as density functional theory, is used to obtain the ground state energy levels of the helium atom along with the Hartree-Fock method.


The Hamiltonian of helium is given by

H\psi(\vec{r}_1,\, \vec{r}_2) = \Bigg[\sum_{i=1,2}\Bigg(-\frac{\hbar^2}{2\mu} \nabla^2_{r_i} -\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}} \Bigg]\psi(\vec{r}_1,\, \vec{r}_2)

where \mu = \frac{mM}{m+M} is the reduced mass of an electron with respect to the nucleus and r_{12} = |\vec{r_1} - \vec{r_2}| . Consider M = \infty so that \mu = m and the mass polarization term \frac{\hbar^2}{M} \nabla_{r_1} \cdot \nabla_{r_2} disappear. The Hamiltonian in atomic units (a.u.) for simplicity is given by

H\psi(\vec{r}_1,\, \vec{r}_2) = \Bigg[-\frac{1}{2}\nabla^2_{r_1} - \frac{1}{2}\nabla^2_{r_2} - \frac{Z}{r_1} - \frac{Z}{r_2} + \frac{1}{r_{12}}\Bigg]\psi(\vec{r}_1,\, \vec{r}_2).

The presence of the electron-electron interaction term 1/r12, makes this equation non separable. This means that \psi_0(\vec{r}_1,\, \vec{r}_2) can't be written as a single product of one-electron wave functions. This means that the wave function is entangled. Measurements cannot be made on on one particle without affecting the other. This is dealt with in the Hartree-Fock and Thomas-Fermi approximations.

Hartree-Fock Method

The Hartree-Fock method is used for a variety of atomic systems. However it is just an approximation, and there are more accurate and efficient methods used today to solve atomic systems. The "many-body problem" for helium and other few electron systems can be solved quite accurately. For example the ground state of helium is known to fifteen digits. In Hartree-Fock theory, the electrons are assumed to move in a potential created by the nucleus and the other electrons. The Hamiltonian for helium with 2 electrons can be written as a sum of the Hamiltonians for each electron:

H = \sum_{i=1}^2 h(i) = H_0 + H^'

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 perturbation term:

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

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

H_0 = \hat{h}_1 + \hat{h}_2


\hat{h}_i = -\frac{1}{2} \nabla_{r_i}^2 - \frac{Z}{r_i}, i=1,2

En1, the energy eigenvalues and \psi_{n,l,m}(\vec{r}_i) , the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy eigenvalues and the normalized eigenfunctions. So:

\hat{h}_i \psi_{n,l,m}(\vec{r_i}) = E_{n_1} \psi_{n,l,m}(\vec{r_i})


E_{n_1} = - \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 wave function will reduce to the 'zero-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 separable and the eigenfunctions can 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 corresponding energies are (in a.u.):

E^{(0)}_{n_1,n_2} = 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 wave function

\psi^{(0)}(\vec{r}_2, \vec{r}_1) = \psi_{n_2,l_2,m_2}(\vec{r}_1) \psi_{n_1,l_1,m_1}(\vec{r}_2)

An exchange of electron labels corresponds to the same energy E^{(0)}_{n_1,n_2} . This particular case of degeneracy with respect to exchange of electron labels is called exchange degeneracy. The exact spatial wave functions of two-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 two electrons. The proper wave function then must be composed of the symmetric (+) and antisymmetric(-) linear combinations:

\psi^{(0)}_\pm(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2}} [\psi_{n_1,l_1,m_1}(\vec{r}_1) \psi_{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^{(0)}_\pm . In order to get this wave function into a single product of one-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 \psi^{(0)}_{-} will vanish, in agreement with the original formulation of the Pauli exclusion principle, in which two electrons cannot be in the same state. Therefor the wave function for helium can be written as

\psi^{(0)}_0(\vec{r}_1, \vec{r}_2) = \psi_1(\vec{r_1}) \psi_1(\vec{r_2}) = \frac{Z^3}{\pi} e^{-Z(r_1 + r_2)}

Where \psi_1 and \psi_2 use the wave functions for the hydrogen Hamiltonian. For helium, Z = 2 from

E^{(0)}_0 = E^{(0)}_{n_1=1,\, n_2=1} = -Z^2 \text{ a.u.}

where E^{(0)}_0 = -4 a.u. which is approximately -108.8 eV, which corresponds to an ionization potential V_P^{(0)} = 2 a.u. ( \simeq 54.4 eV). The experimental values are E _0 = -2.90 a.u. ( \simeq -79.0 eV) and V_p = .90 a.u. ( \simeq 24.6 eV).

The energy that we obtained is too low because the repulsion term between the electrons was ignored, whose affect is to raise the energy levels. As Z gets bigger, our approach should yield better results, since the electron-electron repulsion term will get smaller.

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 w

Atomic weight

Atomic weight (symbol: A) is a dimensionlessphysical quantity, the ratio of the average mass of atoms of an element (from a given source) to 1/12 of the mass of an atom of carbon-12 (known as the unified atomic mass unit). The term is usually used, without further qualification, to refer to the standard atomic weights published at regular intervals by the International Union of Pure and Applied Chemistry (IUPAC) and which are intended to be applicable to normal laboratory materials. These standard atomic weights are reprinted in a wide variety of textbooks, commercial catalogues, wallcharts etc., and in the table below. The fact "relative atomic mass of the element" may also be used to describe this physical quantity, and indeed the continued use of the term "atomic weight" has attracted considerable controversy since at least the 1960s (see below).

Atomic weights, unlike atomic masses (the masses of individual atoms), are not physical constants and vary from sample to sample. Nevertheless, they are sufficiently constant in "normal" samples to be of fundamental importance in chemistry.


The IUPAC definition of atomic weight is:

An atomic weight (relative atomic mass) of an element from a specified source is the ratio of the average mass per atom of the element to 1/12 of the mass of an atom of C.

The definition deliberately specifies "An atomic weight…", as an element will have different atomic weights depending on the source. For example, boron from Turkey has a lower atomic weight than boron from California, because of its different isotopic composition. Nevertheless, given the cost and difficulty of isotope analysis, it is usual to use the tabulated values of standard atomic weights which are ubiquitous in chemical laboratories.

Naming controversy

The use of the name "atomic weight" has attracted a great deal of controversy among scientists. Objectors to the name usually prefer the term "relative atomic mass" (not to be confused with atomic mass). The basic objection is that atomic weight is not a weight, that is the force exerted on an object in a gravitational field, measured in units of force such as the newton.

In reply, supporters of the term "atomic weight" point out (among other arguments) that

  • the name has been in continuous use for the same quantity since it was first conceptualized in 1808;
  • for most of that time, atomic weights really were measured by weighing (that is by gravimetric analysis) and that the name of a physical quantity should not change simply because the method of its determination has changed;
  • the term "relative atomic mass" should be reserved for the mass of a specific nuclide (or isotope), while "atomic weight" be used for the weighted mean of the atomic masses over all the atoms in the sample;
  • it is not uncommon to have misleading names of physical quantities which are retained for historical reasons, such as

It could be added that atomic weight is often not truly "atomic" either, as it does not correspond to the property of any individual atom. The same argument could be made against "relative atomic mass" used in this sense.

Determination of atomic weight

Modern atomic weights are calculated from measured values of atomic mass (for each nuclide) and isotopic composition. Highly accurate atomic masses are available for virtually all non-radioactive nuclides, but isotopic compositions are both harder to measure to high precision and more subject to variation between samples. For this reason, the atomic weights of the twenty-two mononuclidic elements are known to especially high accuracy – an uncertainty of only one part in 38 million in the case of fluorine, a precision which is greater than the current best value for the Avogadro constant (one part in 20 million).

The calculation is exemplified for silicon, whose atomic weight is especially important in metrology. Silicon exists in nature as a mixture of three isotopes: Si, Si and Si. The atomic masses of these nuclides are known to a precision of one part in 14 billion for Si and about one part in one billion for the others. However the range of natural abundance for the isotopes is such that the standard abundance can only be given to about ±0.001% (see table). The calculation is

A(Si) = (27.97693 × 0.922297) + (28.97649 × 0.046832) + (29.97377 × 0.030872) = 28.0854

The estimation of the uncertainty is complicated, especially as the sample distribution is not necessarily symmetrical: the IUPAC standard atomic weights are quoted with estimated symmetrical uncertainties, and the value for silicon is 28.0855(3). Th

From Encyclopedia

Atomic Structure Atomic Structure

The ancient Greek philosophers Leucippus and Democritus believed that atoms existed, but they had no idea as to their nature. Centuries later, in 1803, the English chemist John Dalton, guided by the experimental fact that chemical elements cannot be decomposed chemically, was led to formulate his atomic theory. Dalton's atomic theory was based on the assumption that atoms are tiny indivisible entities, with each chemical element consisting of its own characteristic atoms.✶ ✶See Atoms article for further discussion of Dalton's atomic theory. The atom is now known to consist of three primary particles: protons, neutrons, and electrons, which make up the atoms of all matter. A series of experimental facts established the validity of the model. Radioactivity played an important part. Marie Curie suggested, in 1899, that when atoms disintegrate, they contradict Dalton's idea that atoms are indivisible. There must then be something smaller than the atom (subatomic particles) of which atoms were composed. Long before that, Michael Faraday's electrolysis experiments and laws suggested that, just as an atom is the fundamental particle of an element, a fundamental particle for electricity must exist. The "particle" of electricity was given the name electron. Experiments with cathode-ray tubes, conducted by the British physicist Joseph John Thomson, proved the existence of the electron and obtained the charge-to-mass ratio for it. The experiments suggested that electrons are present in all kinds of matter and that they presumably exist in all atoms of all elements. Efforts were then turned to measuring the charge on the electron, and these were eventually successful by the American physicist Robert Andrews Millikan through the famous oil drop experiment. The study of the so-called canal rays by the German physicist Eugen Goldstein, observed in a special cathode-ray tube with a perforated cathode, let to the recognition in 1902 that these rays were positively charged particles (protons ). Finally, years later in 1932 the British physicist James Chadwick discovered another particle in the nucleus that had no charge, and for this reason was named neutron. As a physical chemist, George Stoney made significant contributions to our understanding of molecular motion. However, this Irish scientist is better known for assigning a name to negative atomic charges, electrons, while addressing the Royal Society of Dublin in 1891. —Valerie Borek Joseph John Thomson had supposed that an atom was a uniform sphere of positively charged matter within which electrons were circulating (the "plum-pudding" model). Then, around the year 1910, Ernest Ruthorford (who had discovered earlier that alpha rays consisted of positively charged particles having the mass of helium atoms) was led to the following model for the atom: Protons and neutrons exist in a very small nucleus, which means that the tiny nucleus contains all the positive charge and most of the mass of the atom, while negatively charged electrons surround the nucleus and occupy most of the volume of the atom. In formulating his model, Rutherford was assisted by Hans Geiger and Ernest Marsden, who found that when alpha particles hit a thin gold foil, almost all passed straight through, but very few (only 1 in 20,000) were deflected at large angles, with some coming straight back. Rutherford remarked later that it was as if you fired a 15-inch artillery shell at a sheet of paper and it bounced back and hit you. The deflected particles suggested that the atom has a very tiny nucleus that is extremely dense and positive in charge. Also working with Rutherford was Henry G. J. Moseley who, in 1913, performed an important experiment. When various metals were bombarded with electrons in a cathode-ray tube, they emitted X rays, the wavelengths of which were related to the nuclear charge of the metal atoms. This led to the law of chemical periodicity, which provided refinement of the periodic table introduced by Mendeleev in 1869. According to this law, all atoms of an element have the same number of protons in the nucleus. It is called the atomic number and is given the symbol Z. Hydrogen is the simplest element and has Z = 1. Through Rutherford's work it was known that that electrons are arranged in the space surrounding the atomic nucleus. A planetary model of the atom, with the electrons moving in circular orbits around the nucleus seemed an acceptable model. However, such a "dynamic model" violated the laws of classical electrodynamics, according to which a charged particle, such as an electron, moving in the positive electric field of the nucleus, should lose energy by radiation and eventually spiral into the nucleus. To solve this contradiction, in 1913, the Danish physicist Neils Bohr (then studying under Rutherford) postulated that the electron orbiting the nucleus could move only in certain orbits, having in each a certain "quantized" energy. It turns out that the colors in fireworks would help prove him right. The colorful lights of fireworks are emitted by "excited" atoms; that is, by atoms that have absorbed extra energy. Light consists of electromagnetic waves, each (monochromatic) color with a characteristic wavelength λ and frequency v. Frequency is related to energy E through the famous Planck equation, E = hν, where h is Planck's constant (6.6256 x 10−34 J s). Note that white light, such as sunlight, is a mixture of light of all colors, so it does not have a characteristic wavelength. For this reason we say that white light has a "continuous spectrum." On the other hand, excited atoms emit a "line spectrum" consisting of a set of monochromatic visible radiations. Each element has a characteristic line spectrum that can be used to identify the element. Note that line emission spectra can also be obtained by heating a salt of a metal with a flame. For instance, common salt (sodium chloride) provides a strong yellow light to the flame coming from excited sodium, while copper salts emit a blue-green light and lithium salts a red light. The colors of fireworks are due to this phenomenon. Scientists in the late nineteenth century tried to quantify the line spectra of the elements. In 1885 the Swedish school teacher Johann Balmer discovered a series of lines in the visible spectrum of hydrogen, the wavelengths of which could be related with a simple equation: in which λ is wavelength, k is constant, a = 2, and b = 3, 4, 5, … This group of lines was called the Balmer series. For the red line b = 3, for the green line b = 4, and for the blue line b = 5. Similar series were further discovered: in the infrared region, the Paschen series (with a = 3 and b = 4, 5 … in the above equation), and much later in the ultraviolet region, the Lyman series (with a = 1 and b = 2, 3 …). In 1896 the Swedish spectroscopist Johannes Rydberg developed a general equation that allowed the calculation of the wavelength of the red, green, and blue lines in the atomic spectrum of hydrogen: where nL is the number of the lower energy level to which an electron falls and nH is the number of the higher energy level from which it falls. R is called the Rydberg constant (1.0974 x 10−7 m−1). R was later shown to be 2π 2me 4Z2/h 3c, where m is the mass of the electron, e is its charge, Z is the atomic number, h is Planck's constant, and c is the speed of light. As noted earlier, Bohr had suggested the quantization of Ruthford's model of the atom. Although he was not aware of the work of Balmer and Paschen when he wrote the first version of his 1913 article, he had incorporated Planck's constant h into his model, which turned out to be an important decision. Bohr assumed that the absorption or emission of radiation can occur only by "jumps" of the electron from one stationary orbit to another. (See Figure 1.) The energy differences between two such allowed orbits then provided the characteristic frequencies of the emitted light. ΔE = E n1 − E n2 = hν Planck's constant h was named by Bohr the "quantum of action." Bohr's theory was in close agreement with many experimental facts regarding one-electron atoms (the hydrogen

From Yahoo Answers

Question:Is '6.02 times ten to the power of 23 chemical units of a pure substance' right, or is there some standard test question answer? Thanks - thought the other was a little complicated!

Answers:"A mole of a substance is the mass of a substance that has the same number of particles as there are atoms in 12g of carbon 12" Im doing AS chemistry and I was also confused with the definition. I asked my teacher and he said just think of it as the mass of any substance that has the same number of atoms relative to a scale where 12g of carbon is 12. You can write down Avogadro's number, but he said it wasn't necessary unless it specifies. Hope this makes things clearer.

Question:Explain how the existence of electron energy levels in atoms give rise to line spectra, which may be emission or absorption spectra? Why is there dark lines in the spectrum? Please explain the difference between absorption, emission and continuous spectrum. What leads to continuous and line spectra?

Answers:Because of the energy levels having definite energies, transitions from one to another are always the same energy. This means there are some energies that there are no transitions corresponding to them, thus there are colors that are not possible to emit from a given element. Keep in mind that the element can absorb or emit these definite energies--so when a continuous spectrum--that's ALL colors passes through a given gas, ONLY these energies are absorbed, thus there are black bands which correspond exactly to the emission spectra . . . the absorbtion and emmission spectra are identical, except for the first you are absorbing the given colors and in the second emitting the colors. Continuous spectra usually come from hot things, black body radiation, whereas line spectra come from the electron energy transitions already discussed.

Question:what does it say about the proton+ electron's combined energy level's Is it that there combined energy is greater than the last possible energy level, which on the question is E(7)= - 0.28eV? or E(infinity)= 0e V? cheers by continuum i mean its not in an energy level- its ionised.

Answers:If the electron is no longer in a bound state, where the energy levels of the atom are the discrete values E_n = E_1 / n^2 < 0 (E_1 = - alpha^2 m c^2 / 2, with alpha the fine structure constant, m the electron mass, and c the speed of light), then it is in a scattering state, where the energy can take on any positive value. So, the answer to your question is E > 0.

Question:the definition is: a structure above the cellular level. What is the word for this meaning?

Answers:the organization for complex organisms goes as follows: ecosystems, organisms, organs systems, organs, tissues, cells, molecules, and atoms. any of these, larger than a cell, would answer your question

From Youtube

Atomic Clock :Atomic ClockThe accuracy of time measurement is one of the major problems throughout the whole history of mankind. As the technology developed, the requirements for clock accuracy grew. Clocks have passed a long way of evolution: sundial, water, sand, mechanical, quartz, and finally, atomic. Atomic, or as it is sometimes called molecular or quantum, clock is a device for measuring time, which uses the natural oscillations of atoms or molecules as a batch process. Atomic clocks have the greatest importance for navigation. Determining positions of spacecrafts, satellites, ballistic missiles, aircrafts, submarines, and vehicular traffic in an automatic mode using satellite communication (GPS, GLONASS) are unthinkable without the atomic clock. Since 1967, the international system of units defines one second as a certain number of periods of the electromagnetic radiation generated during the transition of cesium-133 atom between two levels of the ground state. According to this definition, an atom of cesium-133 is a standard to measure time and frequency. The accuracy of determining the second determines the accuracy of all other major units, such as, for example, a volt or a meter, which contain "second" in their definition. Cesium clocks have been considered the most accurate for the last 50 years. A cesium clock will face 1 second error no earlier than in 70-80 million years. But scientists continue to work on improving atomic clocks. The latest achievement belongs to the ...

2 Atoms In A Molecule- A Level Video :An Music Video made for my A Levels! Hope you like it! (With my finger as an extra! Lol!) :)