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Visualizing Complex-Valued Molecular Orbitals Rachael Al-Saadon, Toru Shiozaki, and Gerald Knizia J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01134 • Publication Date (Web): 22 Mar 2019 Downloaded from http://pubs.acs.org on April 2, 2019
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The Journal of Physical Chemistry
Visualizing Complex-Valued Molecular Orbitals Rachael Al-Saadon,† Toru Shiozaki,∗,† and Gerald Knizia∗,‡ †Department of Chemistry, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208, USA. ‡Department of Chemistry, The Pennsylvania State University, 401A Chemistry Building, University Park, PA 16802, USA. E-mail:
[email protected];
[email protected] Abstract We report an implementation of a program for visualizing complex-valued molecular orbitals. The orbital phase information is encoded on each of the vertices of triangle meshes using the standard color wheel. Using this program, we visualized the molecular orbitals for systems with spin–orbit couplings, external magnetic fields, and complex absorbing potentials. Our work has not only created visually attractive pictures, but also clearly demonstrated that the phases of the complex-valued molecular orbitals carry rich chemical and physical information of the system, which has often been unnoticed or overlooked.
1 Introduction The phase of molecular orbitals is fundamentally important in predicting and understanding chemical reactions. One of the earliest examples in the literature is the demonstration that the selectivity of the Diels–Alder reactions can be explained by the phases of the highest-occupied and lowest-unoccupied molecular orbitals. 1 Molecular orbitals have been ever since considered as a key descriptor for chemical reaction mechanisms. Many of the research articles in computational 1
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chemistry thus report the graphic pictures of molecular orbitals, which are often obtained by the standard visualization softwares such as Molden, 2 Avogadro, 3 IQMol, 4 IboView 5,6 to name a few. Visualization of molecular orbitals and associated properties has been thus far limited to that of real-valued orbitals due to the lack of implementation (see, however, Ref. 7 that we became aware of after submission of this manuscript), even though it is widely known that extension to complexvalued orbitals is conceptionally trivial. This is partly because the software for real-valued orbitals does suffice for traditional electronic structure simulations in which the non-relativistic timeindependent Schr¨odinger equation is solved for bound states with open boundary conditions, owing to the fact that the Hamiltonian for such systems is real and the phase of the orbitals can be chosen to be either +1 or −1. There are, however, various situations in which the molecular orbitals become complex, including simulation of systems with relativistic 8 /magnetic 9,10 contributions, that with periodic 11 /absorbing 12,13 boundary conditions, and that with explicit time dependence. 14 Another example is complex generalized Hartree–Fock wave functions that can describe non-coplaner spin polarization. 15,16 In these cases, complex-valued orbitals are often presented as orbital densities or using their real and imaginary parts seperately. Since computational tools for such systems are becoming widely available in recent years, it is of great importance to be able to visualize such orbitals directly. In this work, we have developed a new computer program that allows for visualizing complexvalued molecular orbitals (and other properties). We apply this program to systems under an external magnetic field, systems with spin–orbit coupling, and systems with the absorbing boundary condition. Numerical examples will be shown to demonstrate surprisingly rich information that is pertained in the phase of those orbitals.
2 Technical details To compute the isosurface of a complex-valued molecular orbital, we first take the norm of the orbitals and reuse the components in the IboView programs 5,6 that have been written for real-
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The Journal of Physical Chemistry
valued molecular orbitals. To do this, we simply evaluate the complex orbitals in real space and take the norm at each point: X |ψi (r)| = Cµi φµ (r) µ
(1)
where ψi is an i-th molecular orbital, φµ are basis functions that are real valued, and Cµi are the molecular-orbital coefficients that are complex. We use the so-called Marching Cubes algorithm 17 for computing the triangular (polygon) meshes of the orbital isosurface. Following the default setting with IboView, the isosurfaces are constructed such that 80% of the molecular orbitals are encapsulated within the surface. The normals of the isosurfaces are computed exactly from the derivative of the wave functions at each of the vertices. This has also been implemented in IboView and reused in this work. 5,6 The derivative of the norm can be computed as ∂w |ψi (r)| = Re[∂w ψi (r)] cos θ(r) + Im[∂w ψi (r)] sin θ(r), X ∂w ψi (r) = Cµi ∂w φµ (r),
(2a) (2b)
µ
θ(r) = arg[ψi (r)],
(2c)
where ∂w = ∂/∂w and w = x, y, and z. Subsequently, we calculate the phase angle θ of the molecular orbital at each of the vertices of the triangular meshes. The phase angle is then converted to a color code based on the RGB color
Figure 1: Color wheel that represents the phases.
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wheel that maps the hue of the colors to [0, 2π], as shown in Fig. 1, or, more quantitatively, −3 ≤ p < −2
R=1
G = p+3 B=0
−2 ≤ p < −1
R = −1 − p G = 1
B=0
−1 ≤ p < 0
R=0
G=1
B= p+1
0≤p
