The Frontier Orbital Phase Angles: Novel QSAR Descriptors for Benzene Derivatives, Applied to Phenylalkylamine Hallucinogens
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The Frontier Orbital Phase Angles: Novel QSAR Descriptors for Benzene Derivatives, Applied to Phenylalkylamine Hallucinogens Brian W. Clare* Division of Science, Murdoch University, Murdoch, W.A. 6150, Australia Received March 9, 1998 A new empirical electronic descriptor, obtained from a molecular orbital calculation and applicable to benzene derivatives, is proposed. It is shown that this descriptor, the frontier orbital phase angle, correlates very strongly with the pharmacological activity in humans of a large series of hallucinogenic phenethylamines. In the largest QSAR study on such hal- lucinogens yet reported, it is demonstrated that the phase of mixing of degenerate frontier orbitals of benzene to form the frontier orbitals of the drug results in the best electronic descriptor yet found for hallucinogenic activity in phenylalkylamines. Introduction In the course of investigations of the quantitative structure-activity relationships (QSAR) of phenylalkyl- amine hallucinogen activity with data produced by quantum chemical calculations1 on the drug molecules, schematic diagrams of the frontier orbitals of the molecules were made. Figure 1 shows a pair of such diagrams for two hallucinogens: mescaline, a drug of relatively low activity, and the much more potent drug known as DOM. A plot is made for both the highest occupied (HOMO) and lowest unoccupied (LUMO) mo- lecular orbitals by running a molecular orbital calcula- tion with the molecule oriented with the Z axis normal to the plane of the benzene ring. The molecule is then sketched in the X-Y plane, and circles are drawn on each atom, with a radius proportional to the coefficient of the atomic pz orbital in the molecular orbital expan- sion. The circles are filled or not, depending on the sign of the coefficient. It can be seen that the plots look rather different, and 2 humans have been totally discontinued, leaving only a it was observed that this difference carried over to other relatively small amount of work done in animals using drugs. To some degree, highly active drugs gave plots the drug discrimination paradigm. such as those for DOM, while drugs of low activity tended to give plots looking more like those for mesca- It is obviously difficult to quantify the hallucinogenic line. This paper represents an attempt to quantify and effects of drugs from the subjective reports of individu- analyze these differences in a much larger series. The als. Additionally, variation between individuals is results may well be applicable to other series of drugs significant, and it is more difficult to compare results based on benzene. for two different people than for two drugs in the same person. As these subjective effects are the primary Hallucinogens of the phenethylamine, tryptamine, concern, there is no way around this problem. Even if and ergoline classes are substances that produce char- a quantifiable effect were found that correlated with the acteristic changes in human consciousness, giving rise hallucinogenic effect in humans, in either humans or to a variety of abnormal phenomena.3,4 While the three other species, it could only be used as long as the classes as a whole produce broadly similar effects, there correlation held. Thus all of the data, however obtained, are individual differences between members, and the must ultimately be referred back to subjective reports, effects can be studied satisfactorily only in humans. The obtained from humans. Limitations on the quantifi- drugs are of great importance for the light they throw ability of the hallucinogen experience itself are acknowl- on normal and abnormal mental function, and they are edged, but it is possible to estimate equivalent doses. used in pharmacological preparations for mass screen- Even at best, effective doses are not comparable between ing of new drugs for antipsychotic properties. Unfor- species. tunately, because of perceptions of abuse, studies in A primary concern in this work is standardization. * Also Department of Chemistry, The University of Western Aus- For the purposes of this study, it was decided to rely tralia, Nedlands, W.A. 6009, Australia. entirely on the very extensive human data collected by 3846 Journal of Medicinal Chemistry, 1998, Vol. 41, No. 20 Clare Figure 2. Representation of degenerate orbital pair showing Figure 3. Degenerate HOMO and LUMO frontier orbital phase angle Θ. plots of benzene, P-like (left) and Q-like (right). A. T. Shulgin and co-workers.5,6 This includes data on orbitals themselves are precisely defined, even in the nearly every known phenylalkylamine hallucinogen. As most approximate calculation. the drugs were tried and compared by the same small Any linear combination of the HOMO pair, CpPH + group of experienced observers, the data on different CqQH, is an equally valid HOMO, where Cp and Cq are drugs are as comparable as it is possible for them to any constants. It is convenient to normalize Cp and Cq 2 2 be. The data itself consists of effective dose: the mean so that Cp + Cq ) 1. This leads to the introduction of of the threshold dose and dose required to produce the the angular parameter ΘH, so that the HOMO becomes full effect of the drug. Limitations on the quality of both cos ΘHPH + sin ΘHQH, and the other orthogonal linear the data itself and its correlation with physical quanti- component is -sin ΘHPH + cos ΘHQH.IfΘH is zero, the ties may be assessed solely on the basis of the statistics first of these combinations is simply PH and will be of the data fitting. referred to as P-like, and if ΘH is 90° it is QH, and will be referred to as Q-like. The angular parameter ΘH Theory describes the phase of the degenerate HOMO orbitals A standard self-consistent field (SCF) calculation on as they mix. While the orbitals are degenerate, ΘH has benzene by any ab initio or semiempirical method gives no physical significance. When the degeneracy is lifted, six orbitals of π symmetry.7 Two of these are degenerate however, it describes the orientation, shape, and sym- and constitute the HOMO. Another pair form the metry of the resulting orbital, such as that shown in degenerate LUMO. The HOMO and LUMO together Figure 2B. A similar argument may be applied to the comprise the frontier orbitals of the molecule. When LUMO. The orientations of P and Q for benzene given the benzene molecule is oriented with its plane normal above, and hence the zeros of ΘH and ΘL were those to the Z axis, the π orbitals take a particularly simple obtained in a particular AM1 calculation for benzene form. For all six π orbitals, the coefficients of the carbon and are arbitrary, and any other orientation could 2px and 2py orbitals and the carbon 2s and hydrogen 1s equally have been chosen as the reference. Where P, orbitals are zero, and the orbitals may be represented Q, and Θ are used without subscripts, they may be by six-component eigenvectors of a matrix, giving the taken to refer to both HOMO and LUMO. coefficients of the carbon 2pz orbitals. They may be A plot of the first of these linear combinations for the regarded as vectors in a six-dimensional space. Being HOMO and LUMO of benzene is shown in Figure 4, orthogonal, they may be pictured as the mutually parts A and B, for various values of ΘH and ΘL. In all perpendicular axes of an ellipsoid, the lengths of which of these plots, the rightmost atom is atom 1 and the are proportional to the eigenvalues (i.e., the orbital atoms are numbered counterclockwise. Collectively, ΘH energies). When two orbital energies are degenerate, and ΘL will be referred to as the frontier orbital phase one mutually perpendicular pair of equal axes defines angles. a circular cross section, as in Figure 2A. Then any other From Figure 4 it can be seen that with increasing Θ mutually perpendicular pair of radii of this circle, such the orbital diagram for both HOMO and LUMO goes ′ ′ as P and Q , that are related to P and Q by rotation from P-like at zero, through transitional forms at 15° through the angle Θ are then an equally acceptable pair to Q-like at 30°, through transitional forms back to of orbitals. For both the degenerate HOMO and LUMO P-like again at 60°, but with a 60° rotation. This then orbitals of benzene, two pairs of orbitals take on a repeats with rotation every 60°. Thus angles of Θ + particularly symmetrical form. These will be designated 60n° are equivalent to Θ except for orientation, for n ) ) x - x - x - x as follows: PH (1/2 3, 1/2 3, 1/ 3, 1/2 3, 0, 1, or 2. For n ) 3, the plot returns to that for n ) 0, x x 1/2 3, 1/ 3) and QH ) (1/2, 1/2, 0, -1/2, -1/2, 0) as with a change of sign. Because an eigenvector multi- the components of the HOMO, and PL ) (-1/2x3, plied by any constant, including -1, is still an eigen- -1/2x3, 1/x3, -1/2x3, -1/2x3, 1/x3) and QL ) (1/2, vector, this change of sign has no physical significance. -1/2, 0, 1/2, -1/2, 0) as the components of the LUMO. While Θ should not be thought of as an angle in the These orbitals are plotted in Figure 3 according to the three-dimensional space of the molecule, its conse- scheme described above for mescaline and DOM. At quences in that space are apparent from the above. first sight, it may appear that the carbon atoms of the For benzene itself, all of these combinations are benzene ring are not equivalent.