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The 1b 1 MO is a lone pair, while the 3a 1, 1b 2 and 2a 1 MO's can be localized to give two O−H bonds and an in-plane lone pair. [30] This MO treatment of water does not have two equivalent rabbit ear lone pairs. [31] Hydrogen sulfide (H 2 S) too has a C 2v symmetry with 8 valence electrons but the bending angle is only 92°.
In 1928 Eugene Wigner and E.E. Witmer proposed rules to determine the possible term symbols for diatomic molecular states formed by the combination of a pair of atomic states with given atomic term symbols. [4] [5] [6] For example, two like atoms in identical 3 S states can form a diatomic molecule in 1 Σ g +, 3 Σ u +, or 5 Σ g + states.
Molecular orbital theory shows that there are two sets of paired electrons in a degenerate pi bonding set of orbitals. This gives a bond order of 2, meaning that there should exist a double bond between the two carbon atoms in a C 2 molecule. [3] One analysis suggested instead that a quadruple bond exists, [4] an interpretation that was ...
Given any finite list of prime numbers , …,, it will be shown that at least one additional prime number not in this list exists. Let P = p 1 ⋅ p 2 ⋯ p n {\displaystyle P=p_{1}\cdot p_{2}\cdots p_{n}} be the product of all the listed primes and p {\displaystyle p} a prime factor of P + 1 {\displaystyle P+1} , possibly P + 1 {\displaystyle ...
If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case + =). If A , B , C can have a common prime factor then the conjecture is not true; a classic counterexample is 2 10 + 2 10 = 2 11 {\displaystyle 2^{10}+2^{10}=2^{11}} .
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" [2] or "∃ =1". For example, the formal statement
If two numbers are both even or both odd, they have the same parity. By contrast, if one is even and the other odd, they have different parity. The addition, subtraction and multiplication of even and odd integers obey simple rules. The addition or subtraction of two even numbers or of two odd numbers always produces an even number, e.g., 4 + 6 ...
This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. It is convenient at this point (per Trautman 1998 ) to call a triple ( a , b , c ) standard if c > 0 and either ( a , b , c ) are relatively prime or ( a /2, b /2, c /2) are relatively prime with a /2 odd.