Is orbital

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

Another example of the difficulty is offered in figure B3.1.5. Flere we display on the ordinate, for helium s (Is ) state, the probability of finding an electron whose distance from the Fie nucleus is 0.13 A (tlie peak of the Is orbital s density) and whose angular coordinate relative to that of the other electron is plotted on the abscissa. The Fie nucleus is at the origin and the second electron also has a radial coordinate of 0.13 A. As the relative angular coordinate varies away from 0°, the electrons move apart near 0°, the electrons approach one another. Since both electrons have opposite spin in this state, their mutual Coulomb repulsion alone acts to keep them apart. 

Figure B3.1.5. Probability (as a fimction of angle) for finding the seeond eleetron in He when both eleetrons are loeated at the maximum in the Is orbital s probability density. The bottom line is that obtained using a Hylleraas-type fiinetion, and the other related to a highly-eorrelated multieonfigurational wavefLinetion. After [22],...

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

The triplet excited state of H2 is obtained by promoting an electron to a higher-energy molecular orbital. This higher-energy (antibonding) orbital is written and can be considered to arise from two Is orbitals as follows ... 

The final wavefunction stUl contains a large proportion of the Is orbital on the helium atom, but less than was obtained without the two-electron integrals. 

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

Fig. 3.13 Some of the possible combinations of atomic Is orbitals for a 2D square lattice corresponding to different values ofkj and ky. A shaded circle indicates a positive coefficient an open circle corresponds to a negative coefficient.

The Is orbital /i = is correct but not normalized. The normalized function governing the probability of finding an electron at some distance r along a fixed axis measured from the nucleus in units of the Bohr radius oq = 5.292 x 10 " m is... 

What is the probability of finding an electron between 0.6 and 1.2 Bohr radii of the nucleus. Assume the electron to be in the Is orbital of hydrogen. 

Sketch the probability of finding an electron in the 2s orbital of hydrogen at distance r from a hydrogen nucleus as a function of r as a contour map with heavy lines at high probability and light lines at low probability. How does this distribution differ from the Is orbital ... 

In the late 1920s, it was shown that the chemical bond existing between two identical hydrogen atoms in H2 can be described mathematically by taking a linear combination of the Is orbitals [Pg.176]

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is... 

We do not know the orbitals of the electrons either. (An orbital, by the way, is not a ball of fuzz, it is a mathematical function.) We can reasonably assume that the ground-state orbitals of electrons I and 2 are similar but not identical to the Is orbital of hydrogen. The Slater-type orbitals... 

The remainder of the input file gives the basis set. The line, 1 0, specifies the atom center 1 (the only atom in this case) and is terminated by 0. The next line contains a shell type, S for the Is orbital, tells the system that there is 1 primitive Gaussian, and gives the scale factor as 1.0 (unsealed). The next line gives Y = 0.282942 for the Gaussian function and a contiaction coefficient. This is the value of Y, the Gaussian exponential parameter that we found in Computer Project 6-1, Part B. [The precise value for y comes from the closed solution for this problem S/Oir (McWeeny, 1979).] There is only one function, so the contiaction coefficient is 1.0. The line of asterisks tells the system that the input is complete. 

The Gaussian, with r in the exponent, drops off faster than the true Is orbital, which has r in the exponent. The Gaussian is too thin at larger distances r from the nucleus (Fig. 8-2). 

The energy is lower (better) than the STO-IG approximation but not as good as the exact Is orbital. The error has been reduced from 15.1% to 9.1%. 

The reason the inner shell of carbon is represented by 6 primitives in this basis is that the cusp in the Is orbital is difficult to approximate with Gaussians that have no cusp. 

This sum describes the polarization of the Is orbital in terms of functions that have PO symmetry by combining an s orbital and po orbitals, one can form a hybrid-like orbital that is nothing but a distorted Is orbital. The appearance of the excited npo orbitals has... 

The magnitude and "shape" of sueh a mean-field potential is shown below for the Beryllium atom. In this figure, the nueleus is at the origin, and one eleetron is plaeed at a distanee from the nueleus equal to the maximum of the Is orbital s radial probability density (near 0.13 A). The radial eoordinate of the seeond is plotted as the abseissa this seeond eleetron is arbitrarily eonstrained to lie on the line eonneeting the nueleus and the first eleetron (along this direetion, the inter-eleetronie interaetions are largest). On the ordinate, there are two quantities plotted (i) the Self-Consistent Field (SCF) mean-field... 

As proven in Chapter 13.Ill, this two-configuration description of Be s electronic structure is equivalent to a description is which two electrons reside in the Is orbital (with opposite, a and (3 spins) while the other pair reside in 2s-2p hybrid orbitals (more correctly, polarized orbitals) in a manner that instantaneously correlates their motions ... 

As ehemists, mueh of our intuition eoneeming ehemieal bonds is built on simple models introdueed in undergraduate ehemistry eourses. The detailed examination of the H2 moleeule via the valenee bond and moleeular orbital approaehes forms the basis of our thinking about bonding when eonfronted with new systems. Let us examine this model system in further detail to explore the eleetronie states that arise by oeeupying two orbitals (derived from the two Is orbitals on the two hydrogen atoms) with two eleetrons. 

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