Allgemein

Satisfactory Agreement with Experimental Data

and is the concept of exchange. This term is similar to the term direct coulomb, but for traded indices. It is a manifestation of the Pauli exclusion principle and acts in such a way that the electrons of the same spin are separated. This term „exchange“ acts only on electrons with the same spin and comes from the determining Slater form of the wave function. Physically, the effect of exchange is that uniform electrons avoid each other. The term exchange contributes significantly to the complexity of these equations. where (T + V) is the kinetic energy ((T)) or the nuclear attraction energy ((V)). Thus, (epsilon_i) is the mean of kinetic energy plus the coulombic attraction to nuclei for an electron in (phi_i) plus the sum over all occupied spin orbitals in Coulomb (psi), minus the exchange interactions of these electrons with the electron in (phi_i). For example, electron 1 in helium (with (Z=2)), then if (phi_i) is an occupied spin orbital, the term (j = i) ([ J_{i,i} – K_{i,i}]) disappears in the sum above and the remaining terms in the sum represent the coulomb exchange interaction minus (phi_i) with all other occupied spin orbitals (N-1). If (phi_i) is a virtual spinorbital, this cancellation does not occur because the sum above (j) (j = i) does not contain.

So you get the Coulomb-minus exchange interaction of (phi_i) with all (N) of the spin orbitals occupied in (psi). Therefore, the energies of occupied orbitals refer to interactions that are suitable for total electrons (N), while energies of virtual orbitals refer to a system with (N+1) electrons. This difference is very important to understand and keep in mind. Experimental (green dots) and calculated (FBA, black line; JFBA, Blue Line; CFBA, red line) Electron angle distributions for Ee = 6.5 eV, q=0.75 a.u. in the plane as shown in Fig. 1. (a) All Data; (b) azimuthal plane (θe=90°±10°); (c),(d) coplanar geometry (φe=0°±10°). Koopmans` theorem states that the first ionization energy is equal to the negative of the orbital energy of the busiest molecular orbital. Therefore, the ionization energy required to create a cation and a detached electron is represented by the removal of an electron from an orbital without altering the wave functions of the other electrons. This is called the „frozen orbital approximation“. Consider the following pattern of detachment or fixation of an electron in an electronic system (N). The Hartree–Fock equations deal precisely with exchange; However, the equations neglect more detailed correlations due to multi-body interactions.

The effects of electronic correlations should not be overlooked; In fact, the failure of the hartree-fock theory to successfully integrate correlation leads to one of its most famous failures. Although Hartree`s equations are numerically manageable via the self-coherent field method, it is not surprising that such an approximate approximation does not capture elements of essential physics. The Pauli exclusion principle requires that the multi-body wave function be antisymmetrical to the exchange of any two electron coordinates. B for example these three terms are identical to hartree`s equations with the approach of the product wave function (i.e. the orbital approximation). The fourth term of the equation (ref{2.9}) is not included in the Hartree equations: the Fock operator is a one-electron operator, and solving a Hartree–Fock equation gives the energy and hartree-fock orbital for an electron. For a system with 2N electrons, the variable i goes from 1 to N; that is, there will be an equation for each orbital. This is because the equation (ref{8.7.2}) uses only spatial wave functions. Since the spatial part of an orbital can be used to describe two electrons, each of the energies and wave functions found by solving the equation (ref{8.7.2}) is used to describe two electrons. (a) Projectile moments in the laboratory xy plane, zoned at φplab=0±30° and φplab=180±30° (Gate px) or φplab=90±30° or φplab=−90±30° (Gate py). (b) The experimental distributions of the angle of electrons in the plane C, gated and und.

Electron angle distributions for a fixed energy of Ee = 6.5±3.5 eV and moment transmission of q = 0.75 ±0.25 a.u. (a) Experimental result and (b) theoretical distribution based on FBA calculations. The areas marked A and C correspond to the so-called azimuthal plane and the coplanar geometry. (c) and (d) display 3D representations of contour diagrams (a) and (b). The blue arrow indicates the direction of q and the green arrow indicates the initial axis of the bar (z). The experimental data presented in point (c) shall be reflected at φ=0 to reduce statistical fluctuations. As the extended version on the far right shows, the first term in this equation, (a {H}^0), is the well-known hydrogen-type operator, which takes into account the kinetic energy of an electron and the potential energy of that electron interacting with the nucleus. The following term takes into account the potential energy of an electron in an average field generated by all other electrons in the system. The operators (a {J}) and (a {K}) result from the electron-electron repulsion terms in the complete Hamilton operator for a multi-electron system. These operators include single-electron orbitals as well as electron-electron interaction energy. Within the limits of the HF-Frozen orbital model, ionization potentials (IP) and electron affinities (EA) are given as negatives of occupied or virtual spin orbital energies.

This statement is called Koopmans` theorem; It is often used in quantum chemical calculations as a means of estimating ionization potentials (equation (ref{8.7.8})) and electron affinities (equation (ref{8.7.9})) and often provides qualitatively correct results (i.e. ± 0.5 eV). In general, the Hartree-Fock theory offers an excellent first-order solution (99%) for describing multi-electron systems, but the latter 1% is still too large to quantitatively describe many aspects of chemistry, and more sophisticated approaches are needed. These are discussed elsewhere. where (has {F}) is called the operator Fock and ( {| varphi_i rangle }) are the Hatree-Fock orbitals with the corresponding energies (epsilon_i). The best possible single-electron wave functions, by definition, result in the lowest possible total energy for a multi-electron system used with the full multi-electron Hamilton system to calculate the expected value of the total energy of the system. These wave functions are called Hartree-Fock wave functions and the total energy calculated is the Hartree-Fock energy of the system. Let us derive this result for the case where an electron is added to the spin orbital (N+1^{st}). The energy of the electron determinant (N) is occupied by the spinorbitals (phi_1) to (f_N) As with the Hartree equations, the resolution of the Hartree–Fock equations is mathematically equivalent to the assumption that each electron interacts only with the average charge cloud of the other electrons. This is how electron-electron repulsion is managed.

For this reason, this approach is also known as the self-coherent field approach (SCF). Although the study of ion-atom collisions is a mature field of atomic physics, large discrepancies between experiment and theoretical calculations are still common. Here we present experimental results at high pulse resolution for the unique ionization of helium induced by protons of 1-MeV and compare them to theoretical calculations. The overall agreement is surprisingly good, and even Born`s first approximation gives a good agreement between theory and experience. .