Acute Triangles Research Articles

Spectroscopy of the predissociated C state of Na3

The C state of Na3 has been explored in detail by the methods of resonant two-photon ionization spectroscopy and photofragmentation spectroscopy of a supercooled cluster beam. The photofragmentation spectrum, recorded by depletion of Na3 from the beam, reveals a long progression of bands that had been invisible to the two-photon ionization method, and it appears that ultrafast fragmentation occurs for all levels more than 400 cm−1 above the zero-point level. This is consistent with earlier observations of Na D-line emission following Na3 excitation in this region, assuming the responsible channel is Na2 2X and Na 2P production. The vibronic fine structure of the C band system is complex at lower energies, and yields to a detailed explanation only through consideration of the dynamical Jahn–Teller effect. This analysis demonstrates that the C state has electronic symmetry E″, and is subject to a symmetry-lowering deformation of 180 cm−1, or 1.4 times the frequency of the e′ vibrational mode of D3h Na3. The corresponding minimum energy structure is an obtuse isosceles triangle (65° apex angle), but the barrier to pseudorotation (estimated to be 40 cm−1) is small. Furthermore, the computed vibronic wave functions imply that only the lowest few states are well described within the adiabatic Born–Oppenheimer approximation; at intermediate energies the spectrum is correspondingly very irregular, while at higher energies a near harmonic regularity returns by virtue of the relatively small magnitude of the Jahn–Teller distortion. These quantitative conclusions are intermediate among those pertaining to the other known states of Na3, and, in combination with them, permit extensive comparison with the results of high quality electronic structure computations. The tentative assignment is to the 12E to 22E″ electronic transition (united atom 1S21Px,y to 1S21Dxz,yz).

Nonobtuse triangulation of polygons

We show how to triangulate a polygon without using any obtuse triangles. Such triangulations can be used to discretize partial differential equations in a way that guarantees that the resulting matrix is Stieltjes, a desirable property both for computation and for theoretical analysis. A simple divide-and-conquer approach would fail because adjacent subproblems cannot be solved independently, but this can be overcome by careful subdivision. Overlay a square grid on the polygon, preferably with the polygon vertices at grid points. Choose boundary cells so they can be triangulated without propagating irregular points to adjacent cells. The remaining interior is rectangular and easily triangulated. Small angles can also be avoided in these constructions.

Potential energy surfaces of low-lying states of the aluminum trimer

The potential energy surfaces for the low-lying states of the Al 3 molecule have been investigated by means of Hartree-Fock CI calculations. Four states are candidates for the ground state: an acute isosceles triangle 4B 1, an obtuse isosceles triangle 4A 2, and two nearly equilateral triangles, 2B 1 and 2A 1. The ground state is found to be 4B 1, but because of the very small energy separating the four states they can be considered as nearly degenerate. A partial vibrational analysis has been carried out considering only the totally symmetric modes of the C 2v point group in order to show the different nature of the corresponding potential energy surfaces.

Stability of Cu3: A variational and perturbational configuration-interaction study.

Three-body nonadditive contributions to the interaction energies for different geometrical arrangements of the copper trimer are shown to be decisive in establishing the optimal ${\mathrm{Cu}}_{3}$ structure. Such an optimal geometry is confirmed to be an obtuse triangle but a linear geometry is shown to be also stable, albeit some 7 kcal/mol higher in energy than the obtuse one. The most symmetrical structure (i.e., an equilateral triangle) is less stable than either the linear or the obtuse triangle, but this originates exclusively from its high three-body repulsive contribution to the binding energy.

CAS SCF/CI calculations of low-lying states and potential energy surfaces of Si 3

Complete active space MC SCF and restricted first-order CI calculations of low-lying states of Si 3 ( 1A 1, 3B 2, 1B 2, 3B 1, 1B 1, 3A 2, 1A 2) and potential energy surfaces of the 1A 1 and 3B 2 states are calculated. MRSDCI calculations are carried out for the 1A 1 and 3A' 1(D 3h) states. The ground state of Si 3 is found to be the 1A 1 state with the 3A' 2 (equilateral triangle) state 3.6 kcal mole above the singlet state in a TZ + ld basis set. A second shallow minimum (obtuse triangle, apex angle 160°) and a barrier are found in the potential energy surface of the 3B 2 state. The equilibrium geometry of the 1A 1 state is an isosceles triangle, apex angle 80° while that of the 3B 2 ( 3A' 22) state is an equilateral triangle. The equilateral and isosceles triangular structures are more stable than the corresponding linear structures.

Absolute motion and evolution of the Bouvet triple junction

The absolute motion of the Bouvet triple junction has an important role in its mechanical evolution. At present, the relative velocity vectors of the South American, African and Antarctic plates form an obtuse isosceles triangle, and both the ridge–ridge–ridge (RRR) and ridge–transform–transform (RFF) configurations of the Bouvet are potentially stable. Sclater1 demonstrated that the Bouvet twice alternated between RRR and RFF in the past 20 Myr. The reasons for these changes are unclear, but the analysis of the absolute velocities of the two configurations presented here provides some insight; The motion of the RRR configuration of the Bouvet, with respect to a hotspot reference frame, carries the junction to the north-west (343°) at 14.7 km Myr−1, whereas the RFF form moves to the south-east (157°) at only 5.8 km Myr−1. The RFF configuration is stable only when the relative velocity triangle is precisely isosceles. As the RFF junction migrates away from the isosceles line, it becomes increasingly unstable, until it is forced to switch to RRR. The RRR junction then drifts back to the isosceles line where it reverts to RFF.

The geometric and electronic structure of small copper clusters Cun and Cu+n (n=1–3) by an effective core potential method

Electronic structure calculations have been carried out for the total energy of the ground state of Cun and Cu+n (n=1–3) clusters. The Cu atom is treated as a one-electron system and the effect of the core is approximated by a shape consistent pseudopotential and a semiempirical core-valence, core–core polarization potential. The exchange and correlation energies among the valence electrons are treated by the local-spin-density-functional approximation with and without the self-interaction corrections (SIC). The binding energy and bond distance of Cu2 calculated with SIC are in almost exact agreement with experimental data. The calculated ionization potentials exhibit even–odd oscillations as a function of n. The potential energy surfaces of Cu3 show substantial differences when obtained with and without SIC. The most stable geometry predicted by the latter is an acute triangle with an apex angle of 47° and a long bond length of 5.41 a.u. In terms of a distortion parameter ρ with respect to an equilateral triangle with an equilibrium bond length of 4.78 a.u. we have ρ=0.435 a.u. This is in excellent agreement with the experimental value of ρ=0.472 a.u. The calculated binding energy of Cu3 with respect to fragmentation into Cu atoms is 2.259 eV in comparison with the experimental value of 3.108±0.135 or 3.058±0.151 eV. The 27% error is probably due to the use of local exchange-correlation energies and the somewhat small basis set.

The dissection of a polygon into nearly equilateral triangles

Every polygon can be dissected into acute triangles. In this paper we prove that every polygon, such that the interior angles are at least π/5, can be dissected into triangles with interior angles all less than or equal to 2π/5. We find necessary conditions on the interior angles of the polygon in order to obtain a dissection into triangles with interior angles all ⩽α (where π/3<α<2π/5). The conjecture can be stated that these conditions are also sufficient.

Electronic structure of small copper clusters. II. Localized d hole in excited states and ionized states of Cu2 and Cu3

The electronic structure of Cu2 and Cu3 clusters is investigated by ab initio SCF calculations. The geometry optimization is performed. Because of the Jahn–Teller effects the obtuse and acute isosceles triangle configurations of Cu3 are energetically lower than the regular triangle one. The ‘‘localized’’ orbital which is located at the top or center atom is commonly found through the obtuse, regular, acute, and linear triangle Cu3’s. The excited and ionized states where the localized d electron participates are energetically close to those of (s → s or s*) and (s → ∞) states, which suggests that the electronic structure of Cu3’s is similar to that of bulk. The calculated excitation energies and ionization potentials for these states agree with experiment within error of 1.0 eV. The localized d hole excited states and the ionized states are found in Cu2 as well. The effect of the basis set superposition on Cu2 and Cu3 and the band structure of larger Cu clusters are also discussed.

Semi-empirical valence bond potential energy surfaces for homonuelear alkali trimers

Semi-empirical valence bond surfaces for the homonuclear alkali trimers (M 3) have been obtained by evaluating the Coulomb and exchange integrals in the London equation, respectively, as half the sum and difference of new reliable analytical potential energy curves for the 1Σ g + and 3Σ u + states of the dimers (M 2). Calculations were also carried out using an effective 3Σ u + universal diatomic function which was chosen to reproduce the experimentally known binding energy of Li3. The ground state equilibrium structures of those trimers are predicted to be an obtuse triangle (2 B 2 symmetry) bound with respect to M 2(1Σ g + )+M(2 S) and ordered Li3>Na3>K3>Rb3>Cs3 according to their stabilities. The surfaces are found to be extremely flat with two saddle points, an acute isosceles triangle (2 A 1 symmetry) and a linear symmetric structure (2Σ u + ) lying above but very close to the minimum. The force constants of the 2Σ u + , 2 A 1 and 2 B 2 structures, as well as the fundamental frequencies and zero poin...

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta1,b1,c1,A1 anda2,b2,c2,A2 be the sides and areas of two triangles. Ifa=(a1p+a2p)1/p,b=(b1p+b2p)1/p,c=(c1p+c2p)1/p, and 1≤p≤4, thena, b, c are the sides of a triangle and its area satisfiesAp/2≥A1p/2+A2p/2. If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaAf, which satisfies (4Af/√3)1/2≥f((4A/√3)1/2).

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝ n , with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

A Strategy For Avoiding A Hidden Trap In Trigonometry

The solution of an oblique triangle, given two sides and the included angle, can contain a hidden trap often overlooked by both students and textbooks. If one of the unknown angles is obtuse, it is possible to obtain an erroneous solution for the triangle.

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Electronic structure of the sodium trimer

An ab initio configuration-interaction study of the potential energy surface of Na3 is presented. The ground state is predicted to be an obtuse triangle (2 B 2 symmetry), and to be bound by 35·5 kJ/mole relative to Na2(1Σ g +) + Na(2 S). The surface is found to be extremely flat. Two saddle points, an acute triangle (2 A 1 state) and a linear symmetric conformation (2Σ u + state), lie respectively only 2·5 kJ/mole and 12·5 kJ/mole above the minimum. The isotropic hyperfine coupling constant and first ionization potential calculated at the minimum energy geometry are in good agreement with experiment. A simple molecular orbital model involving only the valence s-orbitals provides an adequate qualitative description of the electronic structure. The low-lying empty p and d-orbitals in Na influence the stability of Na3 mainly through contributions to electron correlation and not through orbital hybridization. The dominant features of the surface are discussed in terms of pseudo-rotation and the Jahn-Teller instability of the equilateral triangle geometry. Theoretical estimates of excitation energies and oscillator strengths to the optically allowed excited states are also presented.

Quasi-Pythagorean Triples for an Oblique Triangle

Quasi-Pythagorean Triples for an Oblique Triangle

Two Notes on Metric Geometry

If $\mu ,\smallint d\mu = 1$, is a signed Borel measure on the unit ball in ${E^3}$, it is shown that ${\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2$ with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.

Two notes on metric geometry

If μ , ∫ d μ = 1 \mu ,\smallint d\mu = 1 , is a signed Borel measure on the unit ball in E 3 {E^3} , it is shown that sup μ ∫ ∫ | p − q | d μ ( p ) d μ ( q ) = 2 {\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2 with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.

The Unrestricted Hartree-Fock Theory of Chemical Reactions. V: The Reaction of Three Hydrogens in Isosceles Triangular Conformations

A UHF theoretical analysis is carried out for the reaction of three hydrogens in iso­ sceles triangular conformations. It is shown that the UHF ground state of the H 8 system consists of· three different phases; an RHF configuration in a region of acute triangles, an ASDW configuration in a region of flat triangles and a TSDW configuration iri a region near equilateral triangles. The potentials of the UHF states are calculated and their interconnection relation is studied. A configurational analysis in terms of the RHF orbitals is carried out for the UHF states at equilateral triangles. It is shown that the TSDW state is a good approximation to ·the full CI ground state. The reason for the missing of the Jahn-Teller cusp in the UHF ground state potential is discussed. Importance of spin flipping instabilities in mechanistic characterization of radical reactions is pointed out. In previous papers,n-5> we have carried out unrestricted Hartree-Fock (UHF) theoretical analyses of the chemical reactions in singlet ground state. We consider, in this paper, the three hydrogen system with isosceles triangular conformations as one of the simplest model system for insertion reactions of free radicals. We show from an analysis of the H 3 system that instability of a ground state configu­ ration and appearance of a new ground state of broken symmetry character are phenomena which may occur not only in the reacting systems with a singlet ground state but also in the systems with a doublet ground state, i.e., in free radical reactions. The UHF ground state of the H 3 system undergoes phase transitions between a restricted Hartree-Fock (RHF), an axial spin density wave (ASDW) and a torsional spin density wave (TSDW) configurations, indicating the presence of a hierarchy of doublet radicals with electronic structures of increas­ ing complexities as in singlet radicals discussed previously.5> A configurational analysis is carried out for the ASDW and the TSDW states at equilateral triangles in order to clarify the nature of the electronic correlation effects incorporated into the UHF states. The reason for the missing of the Jahn-Teller cusp6> in the UHF ground state

An ab initio molecular orbital study of Li 2O 2 and LiO 2H

An ab initio molecular orbital study, using a Gaussian basis set, of the geometries of Li 2O 2 and LiO 2H is presented. Consistent with the experimental data available, Li 2O 2 is found to have a rhombic structure; the O-Li-O configuration in LiO 2H is found to form an acute triangle. The bonding in these species is discussed to terms of Mulliken population analyses.