Haptic Representation of the Atom

Chaim Gingold Research Assistant West Virginia Virtual Environments Laboratory [email protected] Three-dimensional functions that represent atomic orbitals are traditionally difficult for chemistry students to conceptualize. Large sections of undergraduate physical chemistry texts are devoted to breaking apart and simplifying electron density functions so they can be visually represented. Traditional methodologies include skins (isosurfaces and enclosures of certain probabilities), three-dimensional projections (color, contours, slices) and two-dimensional graphs. Preliminary work with the Phantom 3d haptic interface suggests that haptics are an important addition to the chemist's tool set for representing atomic orbitals. With the Phantom, users simply move through real three-dimensional space and perceive the electron density as the force on the Phantom's pen. In our work, the force is proportional to the probability density function for the electron at any point, given by the square of the wavefunction describing a particular atomic orbital (ψ2(r, θ, φ)). Nodes are felt as regions of zero force, increasing ψ 2 values are felt as increasing resistance, and maxima are communicated by haptic "clicks." Previous application of the Phantom to chemistry by Wanger utilized haptics for molecular docking feedback [1]. The present work tackles the altogether different problem of haptically visualizing the probability density functions of individual atomic orbitals. Feedback from chemistry students is positive and interest in continued development of the work is high among chemistry faculty. 1. Background & Problem Atomic orbitals are three-dimensional mathematical functions used by chemists to describe the locations of electrons in atoms. Understanding the distribution of electron density in atoms is critical for the development of major concepts in chemistry including the structure of the periodic table, the formation of bonds between atoms, molecular geometry and chemical reactivity. The central importance of atomic orbitals to the study of chemistry is highlighted by the inclusion of various representations for the functions even in introductory chemistry texts. Unfortunately, the functions that represent atomic orbitals are complicated and difficult to visualize with traditional methods. The oversimplification required to introduce atomic orbitals at the introductory level is followed by hours of work in the third-year undergraduate physical chemistry course, much of it directed at simply understanding the relationship between the functions and their traditional graphical representations. Solution of the Schrodinger equation for the hydrogen atom produces a family of electron wavefunctions denoted by ψ(r, θ, φ). Multiplying a wavefunction by its complex conjugate produces a real function whose value at any point in space represents the probability density for finding an electron at that point. This real function, known casually as ψ2, is an atomic orbital. Atomic orbital functions generate three-dimensional probability density fields that describe where an electron is likely to be found. Alternatively, the three-dimensional probability density field can be said to determine the shape of a particular orbital. For one-electron atoms, the shape of the atom itself is controlled by the shape of the orbital in which the electron is located. Each atomic orbital is characterized by a unique set of three quantum numbers, n, l and ml, that determines its particular mathematical form. Atomic orbitals with l=0 are known as "s" orbitals. When n=1, the function is the 1s orbital; n=2 produces the 2s orbital, and so on. The mathematical functions that correspond to s orbitals consist of a constant, multiplied by a polynomial in r (of degree n-1), multiplied by an exponential decay in r. The 1s and 2s atomic orbital functions, shown below, have spherically symmetric shapes because the functions have no dependence on θ or φ. ψ21s(r, θ, φ) = [N1*e(-Zr/ao)]2 ψ22s(r, θ, φ) = [N2*(2 Zr/ao)*e(-Zr/2ao)]2 In the equations above, N1 and N2 are constants, Z is the (integral) number of protons in the nucleus of the atom and ao is a constant (52.9 picometers) that provides a convenient unit for atomic dimensions. Atomic orbital functions take the general form shown below, where the radial and angular variables are separable. ψ2(r,θ,φ) = R2(r)*Y2(θ,φ) Values of l>0 add angular dependence to the types of functions shown above for the 1s and 2s orbitals, and lead to orbitals with dumbbell (l=1), cloverleaf (l=2) or even more complex shapes. The complexity of the radial dependence of the function also increases for orbitals with higher quantum numbers. Difficulty in conceptualizing the functions increases correspondingly. The major challenge in representing atomic orbital functions arises because each location in 3-dimensional space has an associated value of ψ2. If three dimensions are used to locate a point in space, a fourth dimension is needed to communicate the value of ψ2. Traditional representations of orbital shapes are based on attempts to reduce the dimensionality of the ψ2 functions, and physical interpretations of the apparent shapes vary with the method of representation. Introductory chemistry students presented with traditional images of atomic orbitals typically remain unaware of the limited nature of the representations. These students often develop firm misconceptions that slow the process of understanding the representations later in the curriculum. Furthermore, when this topic is covered in junior-level physical chemistry, the chemically-relevant understanding of orbitals too often gets lost in the maze of subtly different representations used to attempt to convey the shapes. 2. Traditional Representations Figure 1a. Radial dependence of 2s atomic orbital Figure 1b. Radial dependence of 2s atomic orbital Common orbital representation methods involve plotting the radial and angular dependencies separately. A full understanding of the orbital shapes then requires the viewer to synthesize the information from the two types of plots. In Figures 1a and 1b, the value of the electron probability density function for the 2s orbital is plotted on the vertical axis as a function of distance from the nucleus. Figure 1a clearly shows the general decay of electron density from a finite value at the nucleus to very small values at 10*ao. Figure 1b is the same plot, rescaled with a smaller maximum for the vertical axis in order to highlight the radial node at Zr/ao = 2 and the local maximum in electron density at approximately Zr/ao = 4. Figure 2. Angular dependence for 2s orbital Figure 3. Angular dependence for l=1 orbital In Figure 2 the radial dependence is ignored and the value of Y2(θ,φ) for an s orbital is represented as the apparent distance from the origin. Figure 3 shows the same type of plot for an orbital with l=1. While these representations clearly show the angular dependence of the functions, no information about radial features (such as radial nodes) is communicated. The plots look similar to quite different plots in which all three dimensions represent spatial coordinates and the orbital shape is shown by an isosurface, or skin of constant ψ2. The similarity in appearance provides a continual source of confusion for students. Both types of plots also reinforce the incorrect idea that orbitals have a definite outline or edge. Figure 4. Surface plot for 2s orbital In Figure 4, the value of ψ 2 for a 2s orbital is represented by the height on the z-axis, while the x and y dimensions are real spatial dimensions. The nucleus of the atom is located at the center of the xy plane. This figure illustrates electron density features for a slice of the orbital in the x-y plane. A subtle feature is the presence of a radially symmetric bump of electron density at the base of the cone, corresponding to the local maximum shown in Figure 1. A disadvantage of this representation is that many slices are needed to construct a full mental picture of the orbital. The most complete and conceptually accurate representations of atomic orbitals involve the use of color or dot frequency to denote a fourth dimension in a 3d plot. The average electron density in small volume elements is calculated and then represented by color or number of dots. This type of representation is most effective when the image can be rotated. The major drawback is that internal details about electron density are obscured by the outer parts of the image. A navigable three dimensional representation is needed, one which would allow the viewer to fly through the orbital. We achieve this goal by using haptic force to represent a fourth dimension. 3. A Haptic Approach Sensable Technology's Phantom allows the user to move a pen, connected to a mechanical arm, within a 3d workspace on the user's desk. The computer is capable of applying forces to the pen, making full duplex haptic communication between the computer and user possible [2]. The software we developed directly maps the 3-space of the atom into the 3-space of the Phantom's workspace. The electron probability density function drives the force exerted on the user. The user, by moving the Phantom's pen around the workspace, probes different points in the atomic orbital. The Phantom responds by continuously updating the forces acting on the user with the output from the ψ2 function, creating a tangible electron probability density field. On the computer screen, a twodimensional projection of a three-dimensional electron density isosurface provides the user with an additional point of reference for understanding the location of the probe in relation to the nucleus. The magnitude of the force vector is calculated by scaling the output of the ψ function at the location of the probe. The gradient of the probability function governs the direction in which the pen is pushed. So far, we have modeled the spherically symmetric 1s and 2s atomic orbitals. Because the haptic device communicates only a limited range of forces, it was necessary to zoom in on the interesting parts of each orbital to make interesting features haptically appreciable.