The trihydrogen cation

Like the dihydrogen cation, hJ, the trihydrogen cation, H, is known only in the gas phase. Both ions have, however, been found in interstellar space. The structure of is that of an equilateral triangle with the three H-H distances equal to 87.5 pm as compared to 74 pm in H2 or 106 pm in hJ. These distances are so short as to imply the existence of three H-H bonds, though the total number of electrons in the molecule is just two The dissociation energy, defined as the energy of the reaction 

Calculating the derivatives of e with respect to each of the coefficients Ci, C2, and C3, setting the three derivatives equal to zero in order to find the lowest possible value of e, and proceeding as outlined in Section 8.1, we obtain the three secular equations  

Since the three atoms are identical, the three integrals Hi i, i/2,2 and 7/3,3 are equal. Since the three atoms occupy the comers of an equilateral triangle, all the resonance integrals must also be equal 7/1,2 = 2,1 = /fi,3 — 3,1 = 7/2,3 = 7/3,2. If we replace all overlap integrals S by zero, the secular equations simplify to 

These homogeneous linear equations will have non-trivial solutions if the secular determinant is equal to zero  

Since the resonance integrals Hi 2 are negative, the first energy is that of a bonding orbital, the second and third those of two degenerate, antibonding orbitals. 

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Tlie simplest known example of a 3c-2e bond is that in the trihydrogen cation (H3+), the existence of which, in the gas phase, was first demonstrated... 

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