Even a slight increase in molecular size within a simulation can dramatically increase the computational burden. So practitioners often resort to using empirical information, which describes the motion of atoms in terms of classical physics and Newton's Laws, enabling simulations that scale to billions of atoms or millions of chemical compounds. Traditionally, empirical potentials have had to strike a tradeoff between accuracy and transferability. When the many parameters of the potential are finely tuned for one compound, the accuracy decreases on other compounds.

Instead, the Los Alamos team, with the University of North Carolina at Chapel Hill and University of Florida, has developed a machine learning approach called transfer learning that lets them build empirical potentials by learning from data collected about millions of other compounds. The new approach with the machine learning empirical potential can be applied to new molecules in milliseconds, enabling research into a far greater number of compounds over much longer timescales.

The Los Alamos authors acknowledge support of the U. DOE Office of Science. This research in part was done using resources provided by the Open Science Grid which is supported by the National Science Foundation award , and the U. Note: Content may be edited for style and length. Science News. Funding The Los Alamos authors acknowledge support of the U.

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Journal Reference : Justin S. Smith, Benjamin T. Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning. Nature Communications , ; 10 1 DOI: ScienceDaily, 2 July The energy levels only depend on n , the principal quantum number and are given by equation 1. The electron wavefunctions however are different for every different set of quantum numbers. While a derivation of the actual wavefunctions is beyond the scope of this text, a list of the possible quantum numbers is needed for further discussion and is therefore provided in Table 1.

For each principal quantum number n , all smaller positive integers are possible values for the quantum number l. The wave nature of particles allows for the possibility that a particle penetrates a thin barrier even if the particle energy is less than the height of the barrier. This phenomenon is referred to as tunneling. From a classical mechanics point of view, tunneling cannot easily be explained since it would be the equivalent of a ball going through a wall without damaging the wall.

The analogy is further incorrect in that only part of the incident particles tunnel through the barrier, while most are reflected back. A better but still incomplete analogy is that of light penetrating through a thin metal layer. We now calculate the transmission and reflection of an electron through a thin potential barrier with height V 0 and width d as shown in Figure 1.

We now consider the situation where a wave with amplitude A is incident from the left side of the barrier. The general solution in each of the regions is then:. This leads to the following set of homogenous equations:. The full solution is obtained by first solving for E , while A is considered to be known. The other constants can be solved from the equations above. A common approximation assumes that the transmission through the barrier is small so that the e a d term is real an much larger than e - a d.

The transmission probability then equals the square of ratio of the transmitted wave to the incident wave, or:. As an example we now consider tunneling of an electron through a 1 nm wide, 1 eV barrier. The calculated transmission as obtained by solving 1. For an electron energy less than 1 eV one observes significant transmission due to tunneling, which increases exponentially as the energy approached the height of the barrier. A resonant condition is obtained at an electron energy of 1.

## Chapter 1: Review of Modern Physics

The figure clearly illustrates the resonance within the barrier. The probability density function of the forward propagating wave is unity out side the barrier and peaks within the barrier. Out side the barrier, there only exists a forward propagating wave as can be seen from the 90 o phase shift between the real and imaginary part of the wavefunction. Once the energy levels of an atom are known, one can find the electron configurations of the atom, provided the number of electrons occupying each energy level is known.

Electrons are Fermions since they have a half integer spin. They must therefore obey the Pauli exclusion principle. This exclusion principle states that no two Fermions can occupy the same energy level corresponding to a unique set of quantum numbers n , l , m or s. The ground state of an atom is therefore obtained by filling each energy level, starting with the lowest energy, up to the maximum number as allowed by the Pauli exclusion principle. The electronic configuration of the elements of the periodic table Appendix 7 can be constructed using the quantum numbers of the hydrogen atom and the Pauli exclusion principle, starting with the lightest element hydrogen.

Hydrogen contains only one proton and one electron. The orbital quantum number, l , equals zero and is referred to as an s orbital not to be confused with the quantum number for spin, s. The s orbital can accommodate two electrons with opposite spin, but only one is occupied. This leads to the shorthand notation of 1s 1 for the electronic configuration of hydrogen as listed in Table 1. This table also lists the atomic number which equals the number of electrons , the name and symbol, and the electronic configuration of the first 36 elements of the periodic table.

Helium is the second element of the periodic table. For this and all other atoms one still uses the same quantum numbers as for the hydrogen atom. This approach is justified since all atom cores can be treated as a single charged particle, which yields a potential very similar to that of a proton. While the electron energies are no longer the same as for the hydrogen atom, the electron wavefunctions are very similar and can be classified in the same way.

Since helium contains two electrons it can accommodate two electrons in the 1s orbital, hence the notation 1s 2. Since the s orbitals can only accommodate two electrons, this orbital is now completely filled, so that all other atoms will have more than one filled or partially filled orbital.

The two electrons in the helium atom also fill all available orbitals associated with the first principal quantum number, yielding a filled outer shell. Atoms with a filled outer shell are called noble gases, as they are known to be chemically inert. Lithium contains three electrons and therefore has a completely filled 1s orbital and one more electron in the next higher 2s orbital.

The electronic configuration is therefore 1s 2 2s 1 or [He]2s 1 , where [He] refers to the electronic configuration of helium. Beryllium has four electrons, two in the 1s orbital and two in the 2s orbital. The next six atoms also have a completely filled 1s and 2s orbital as well as the remaining number of electrons in the 2p orbitals. Neon has six electrons in the 2p orbitals, thereby completely filling the outer shell of this noble gas. The next eight elements follow the same pattern leading to argon, the third noble gas.

After that the pattern changes as the underlying 3d orbitals of the transition metals scandium through zinc are filled before the 4p orbitals, leading eventually to the fourth noble gas, krypton. Exceptions are chromium and zinc, which have one more electron in the 3d orbital and only one electron in the 4s orbital.

A similar pattern change occurs for the remaining transition metals, where for the lanthanides and actinides the underlying f orbitals are filled first. This section is devoted to some specific quantum structures that are present in semiconductor devices. These are 1 the finite quantum well, a more realistic version of the infinite well as found in quantum well laser diodes, 2 a triangular well, as found in MOSFETs and HEMTs, 3 a quantum well in the presence of an electric field as found in electro-optic modulators based on the quantum confined stark effect and 4 the harmonic oscillator which has a quadratic confining potential.

The finite rectangular quantum well is characterized by zero potential inside the well and a potential V 0 outside the well, as shown in Figure 1. Since we are looking for bound states i. Since the electron is expected to be close to the well, the wavefunction must go to zero as x approaches infinity.

The wavefunctions outside the well can then be simplified to:. We now use a symmetry argument to further simplify the problem and to find the electron energies. The even solution in the well is a cosine function, while the odd function in the well is a sine function. After squaring and multiplying with the ground state energy, E 10 , of an infinite well with width L x , equation 1. One should note that the derivation of equation 1. Starting with a reasonable guess for E n for instance E 10 , provided it is smaller than V 0 the exact value is obtained through iteration.

The lowest value, E n,min , for a specific quantum level, n , is obtained from equation 1. Since both axes are normalized, Figure 1. Equation 1. The same conclusion can be drawn from the graphical solution in Figure 1. We can also find the number of bound states, n max , from equation 1. Since the maximum energy of a bound state equals V 0 one finds:.

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The Airy function is shown in Figure 1. The solution for a triangular well can also be applied to a quantum well with width L x and infinitely high barriers, in the presence of an electric field. The potential and electron wavefunction are shown in the Figure 1. Since the wave function must be zero at both boundaries, these boundaries must coincide with two different zeros of the Airy function. Since the zeros are discreet rather than continuous our solution will only be valid for specific electric fields.

For intermediate values of the electric field we will interpolate linearly. From Figure 1. This expression can be solved for the electric field as:. It can be shown that in the limit for i going to infinity, which corresponds to a zero electric field, the energy levels correspond those on the infinite quantum well.

The energy levels as a function of the electric field are obtained by selecting a value for i and calculating the corresponding field and energy. The resulting energies, normalized to the first energy level in an infinite well, E 10 , are shown in Figure 1. The polynomials which satisfy equation 1. As an example, we present the wavefunctions for the first three bound states in Figure 1. Maximum kinetic energy, K. The energy is plotted versus the inverse of the wavelength of the light.

Example 1. A metal has a workfunction of 4. What is the minimum photon energy in Joule to emit an electron from this metal through the photo-electric effect? What are the photon frequency in Terahertz and the photon wavelength in micrometer? What is the corresponding photon momentum? What is the velocity of a free electron with the same momentum? This also equals The corresponding photon frequency is: The corresponding wavelength equals: The photon momentum, p, is: And the velocity, v, of an electron in vacuum with the same momentum equals Where m0 is the free electron mass.

Spectral density of a blackbody at , , and K versus energy. The spectral density of the sun peaks at a wavelength of nm. If the sun behaves as a black body, what is the temperature of the sun? A wavelength of nm corresponds to a photon energy of: Since the peak of the spectral density occurs at 2. Energy levels and possible electronic transitions in a hydrogen atom.

Solving for the radius of the trajectory one finds the Bohr radius, a 0 :. Potential energy of an infinite well, with width L x. Also indicated are the lowest five energy levels in the well. The figure is calculated for a 10 nm wide well containing an electron with mass m 0. The wavefunctions and the probability density functions have an arbitrary magnitude i. An electron is confined to a 1 micron thin layer of silicon. Assuming that the semiconductor can be adequately described by a one-dimensional quantum well with infinite walls, calculate the lowest possible energy within the material in units of electron volt.

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If the energy is interpreted as the kinetic energy of the electron, what is the corresponding electron velocity? The effective mass of electrons in silicon is 0. The lowest energy in the quantum well equals: The velocity of an electron with this energy equals:. The probability densities are shifted by the corresponding electron energy. Where we assume that the mass within the barrier is the same as outside the barrier.

Transmission across a 1eV barrier. Resonant transmission across a 1eV barrier.

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Shown is the probability density function of the total wavefunction and separately for the reflected wave as well as the real and imaginary part of the wavefunction on both sides and within the barrier. The origin is chosen in the middle of the well. Graphical solution to the finite rectangular quantum well. Solutions are obtained at the intersection of the curves.

Potential and electron wavefunction for an infinitely deep well within an electric field.