Wave functions contour plots

The most popular way for visualizing MOs is the density or wave function contour plot. We can also introduce other quantities that can measure the position and the spatial extension of an LMO. The position of an LMO can be characterized by the so-called orbital centroid, the expectation value of the position of an electron on the given LMO, r, = (pt r (pt). The spatial extension, the size of an LMO can be measured by the dispersion of electron coordinates placed on a given LMO... 

Having considered the Hartree-Fock description of the helium atom, let us now turn to the exact wave function for this system. Figure 7.2 shows, for the exact ground-state wave function, a plot similar to that for the Hartree-Fock wave function in Figure 7.1. As seen from the distortion of the concentric contour lines close to the fixed electron, the amplitude of the free electron now depends on its position relative to the nucleus as well as to the fixed electron. Moreover, a careful comparison of Figures 7.1 and 7.2 reveals that the probability of finding the two electrons close to each other is overestimated at the Hartree-Fock level. Still, the description afforded by the Hartree-Fock model is reasonably accurate, differing from that of the exact wave function only in the details. 

If a iTioleciile is rotated by chan gin g th c position of the viewer (left mouse btiLlon rotation) ih en the moleetile s position in ihetnolee-ular eoorditi ale system h as not ch an ged and anolb er con lour plot can be requested without recotn pu tin g the wave ftinetion. fb at is, m any orbitals can he plotted after a sin gle poin t ah initio or setn i-einpirical calculation,. iti y contour map is available without recotn putation of the wave function. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

If a molecule is rotated by changing the position of the viewer (left mouse button rotation) then the molecule s position in the molecular coordinate system has not changed and another contour plot can be requested without recomputing the wave function. That is, many orbitals can be plotted after a single point ab initio or semi-empirical calculation. Any contour map is available without recomputation of the wave function. 

Figure 2.19. Contour plots of orbitals for N2 in the gas phase and adsorbed on Ni(100). Solid and dashed lines indicate different phases of the wave function. From Ref. [3].

Figure 14. Contour plots of the wave functions for DCO in the ground bending states 03 = 0. The numbers on the right-hand side denote the polyad number N = vi + V2 + 03/2. The quantum numbers above each column, (vi, V2, 113), indicate the assignment if the substantial mixing between the modes were not present. For more details see the text and the original publication [17], For ease of the visualization the relevant potential cut is shown in the upper left comer. (Reprinted, with permission of IOP Publishing, from Ref. 8.)...

Figure 2.2 (a) Contour plot of the wave function of HOMO level for GaAs cluster model... 

The 2p orbitals on carbon also have one node each, but they have a completely different shape. They point mutually at right angles, one each along the three axes, x, y and z. A plot of the wave function for the 2px orbital along the x axis is shown in Fig. 1.10a, and a contour plot of a slice through the orbital is shown in Fig. 1.10b. Scale drawings of p orbitals based on the shapes defined by these functions would clutter up any attempt to analyse their contribution to bonding, and so it is conventional to draw much narrower lobes, as in Fig. 1.10c, and we make a mental... 

Fig. 12.7 Illustration of the rc-electron interaction of with Ni [3]. (Left) Contour plots of 7t orbitals from DFT calculations for in the gas phase and adsorbed on Ni(lOO). Solid and dashed lines indicate different phases of the wave function. Right) Schematic illustration of the k orbital interactions in the allylic configuration of the N -Ni adsorption system in terms of the atomic N2p and Ni34 orbitals. Reprinted with permission from ref [3]. Copyright 2004 Elsevier...

FIGURE 4.27 Wave function for a particle in a square box in selected quantum states. Dimensionless variables are used, (a) Three-dimensional plot for the ground state f nix, y). (b) Contour plot for Pnlx, y). (c) Three-dimensional plot for the first excited state 4 21 (ii. /) (d) Contour plot for 2i(x, y). (e) Three-dimensional plot for the second excited state 22(x, y). (f) Contour plot for 4 22(x. /) ... 

Finally, we plot the wave function for the second excited state 2i(x, y) (see Fig. 4.27e). It has two maxima (positive) and two minima (negative) located at the values 0.25 and 0.75 for x and y. There are two nodal lines, along x = 0.5 and y = 0.5. They divide the x-y plane into quadrants, each of which contains a single maximum (positive) or minimum (negative) value. Make sure that you see how these characteristics trace back to the one-dimensional solutions in Figure 4.24. Figure 4.27f shows the contour plots for y). As the magnitude (absolute... 

The pattern is now clearly apparent. You can easily produce hand sketches, in three dimensions and as contour plots, for any wave function for a particle in a square box. You need only pay attention to the magnitude of the quantum numbers and Uy, track the number of nodes that must appear along the x and y axes, and convert these into nodal lines in the x-y plane. 

Be certain you understand these choices in each image you examine (or create ). These same issues appear in Chapter 5 when we discuss the wave functions for electrons in atoms, called atomic orbitals. Throughout this book, we have taken great care to generate accurate contour plots and isosurfaces for them from computer calculations to guide your thinking about the distribution of electrons in atoms and molecules. 

FIGURE 5.4 Four representations of hydrogen s orbitals, (a) A contour plot of the wave function amplitude for a hydrogen atom in its Is, 2s, and 3s states. The contours identify points at which i//takes on 0.05, 0.1, 0.3, 0.5, 0.7, and 0.9 of its maximum value. Contours with positive phase are shown in red those with negative phase are shown in blue. Nodal contours, where the amplitude of the wave function is zero, are shown in black. They are connected to the nodes in the lower plots by the vertical green lines, (b) The radial wave functions plotted against distance from the nucleus, r. (c) The radial probability density, equal to the square of the radial wave function multiplied by 1. (d) The "size" of the orbitals, as represented by spheres whose radius is the distance at which the probability falls to 0.05 of its maximum value. 

The chemist s sketches, which are typically drawn to emphasize directionality of the sp hybrid orbitals, and a contour plot of the actual shape, are shown in Figure 6.44. Each of these contours can be rotated about the x-y plane to produce a three-dimensional isosurface whose amplitude is chosen to be a specific fraction of the maximum amplitude of the wave function. These isosurfaces demonstrate that sp hybridization causes the amplitude of the boron atom to be pooched out at three equally spaced locations around the equator of the atom (see Fig. 6.42). The 2p orbital is not involved and remains perpendicular to the plane of the sp hybrids. The standard chemist s sketches of the sp hybrid orbitals and a contour plot that displays the exact shape and directionality of each orbital are shown in Figure 6.44. The isosurfaces shown in Figure 6.43 were generated from these contour plots. 

We present quantitative, computer-generated plots of the solutions to the particle-in-a-box models in two and three dimensions and use these examples to introduce contour plots and three-dimensional isosurfaces as tools for visual representation of wave functions. We show our students how to obtain physical insight into quantum behavior from these plots without relying on equations. In the succeeding chapters we expect them to use this skill repeatedly to interpret quantitative plots for more complex cases. 

Fig. 14.6 The /orbitals (a) plots of the angular part of the wave functions of the/orbitals (b) contours of a 4/orbital. Dots indicate maxima in electron density. The lines are drawn for densities which are 10y of maximum. 1(a) From Friedman, H. G. Choppin, G. R. Feurerbacher, D. G. J. Chern. Educ. 1964, 41, 354-358. (b) From Ogryzio, E. A. J. Cfiein. Ediic. 1965, 42, 150-151. Reproduced with permission.)...

The perfect-pairing (PP) orbitals of this wave function clearly show the "lone-pairs" and "bond pairs" which are part of the language of the experimental chemist. This is in contrast to the molecular orbital description or to the GVB description with (7-7T restrictions where the lone pairs and "7T" bonds are not discernable from contour plots of the orbitals (2 ). It is somewhat reassuring that the wave function which gives the lowest variational energy (that of Figures 1 and 2a) also most closely coincides with the experimental chemist s traditional view of the bonding ( 3) ... 

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