For example, the point group for the water molecule is C, with symmetry operations E, C, σ and σ'. Its order is thus 4. Each operation is its own inverse. As an example of closure, a C rotation followed by a σ reflection is seen to be a σ' symmetry operation: σ*C = σ'. (Note that "Operation A followed by B to form C" is written BA = C).
Another example is the ammonia molecule, which is pyramidal and contains a three-fold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an N-H bond and bisects the H-N-H bond angle opposite to that bond. Thus ammonia molecule belongs to the C point group which has order 6: an identity element E, two rotation operations C and C, and three mirror reflections σ, σ' and σ".
The following table contains a list of point groups with representative molecules. The description of structure includes common shapes of molecules based on VSEPR theory.
Symmetry operations can be written in many ways. A good way to write them is by using matrices. For any vector representing a point in Cartesian coordinates, left-multiplying it gives the new place of the point transformed by the symmetry operation. Composition of operations is done by matrix multiplication. In the C example this is:
Although an infinite (going on forever) number of such representations (ways of showing things) exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations. (The irreps span the vector space of the symmetry operations.) Chemists use the irreps to sort the symmetry groups and to talk about their properties.
For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. The tables are square because there are always equal numbers of irreducible representations and groups of symmetry operations.
The table itself is made of characters which show how a particular irreducible representation changes when a particular symmetry operation is applied (put to it). Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But for acting on a general entity (thing), such as a vector or an orbital, this does not have to be what happens. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or -1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation ("symmetric") and -1 denotes a sign change ("asymmetric").
The representations are labeled according to a set of conventions:
The tables also tell the Cartesian basis vectors, rotations about them, and quadratic functions of them transformed by the symmetry operations of the group. The table also shows which irreducible representation transforms in the same way (on the right hand side of the tables). Chemists use this because chemically important orbitals (in particular "p" and "d" orbitals) have the same symmetries as these entities.
The character table for the C symmetry point group is given below:
For example, water (HO) which has the C symmetry described above. The 2"p" orbital of oxygen is oriented perpendicular to the plane of the molecule and switches sign with a C and a σ'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, -1, 1, -1}, corresponding to the B irreducible representation. Similarly, the 2"p" orbital is seen to have the symmetry of the A irreducible representation, 2"p" B, and the 3"d" orbital A. These assignments and others are in the rightmost two columns of the table.
J. Paul Getty
Jean Paul Getty (December 15, 1892 – June 6, 1976) was an American industrialist. He founded the Getty Oil Company. In 1957 "Fortune" magazine named him the richest living American, and the 1966 Guinness Book of Records named him as the world's richest private citizen, worth an estimated $1,200 million. At his death, he was worth more than $2 billion. A book published in 1996 ranked him as the 67th richest American who ever lived. (The book ranked his wealth as a percentage of the United States gross national product.) Despite his wealth, Getty was known for being a miser.
Getty enjoyed collecting art and antiquities. His collection formed the basis of the J. Paul Getty Museum in Los Angeles, California, and over $661 million of his estate was left to the museum after his death. He established the J. Paul Getty Trust in 1953. The trust is the world's wealthiest art institution. It operates the J. Paul Getty Museum, the Getty Foundation, the Getty Research Institute, and the Getty Conservation Institute.
His father George Getty owned a petroleum business in Minneapolis, Minnesota. J. Paul was one of the first people in the world with a fortune estimated at over one billion U.S. dollars.
He enrolled at the University of Southern California, then at University of California, Berkeley before graduating in 1914 from Magdalen College, Oxford with degrees in economics and political science. He spent his summers between studies working on his father's oil fields in Oklahoma. Running his own oil company in Tulsa, he made his first million by 1916. However, in 1917, he announced that he was retiring to become a Los Angeles-based playboy. Although he eventually returned to business, Getty had lost his father's respect. Just before George Franklin Getty died in 1930, he believed that Jean Paul would destroy the family company, and told him so.
After taking a few years off from the money-making grind to enjoy spending his earnings on women, Getty returned to Oklahoma in 1919. During the 1920s he added about $3 million to his already sizable estate. His succession of marriages and divorces (three during the 1920s, five throughout his life) so distressed his father, however, that J. Paul inherited a mere $500,000 of the $10 million the senior Getty left at his death in 1930.
Getty carefully invested his resources during the Great Depression. Getty acquired Pacific Western Oil Corporation. He began the acquisition (completed in 1953) of the Mission Corporation, which included Tidewater Oil and Skelly Oil. In 1967 the billionaire merged these holdings into Getty Oil.
Beginning in 1949, Getty paid Ibn Saud $9.5 million in cash and $1 million a year for a 60-year concession to a tract of barren land near the border of Saudi Arabia and Kuwait. No oil had ever been discovered there. After spending $30 million over four years, oil was discovered there. From 1953 onward, Getty's gamble produced a year, which contributed greatly to the fortune which made him one of the richest people in the world.
Getty learned to speak Arabic which helped his unparalleled expansion into the Middle East. Getty owned the controlling interest in nearly 200 businesses, including Getty Oil. Associates identified his overall wealth at between $2 billion and $4 billion. It did not come easily, perhaps inspiring Getty's widely quoted remark—"The meek shall inherit the earth, but not the mineral rights."
He moved to England in the 1950s and loved English culture (Anglophile). He lived and worked at his 16th-century Tudor estate, Sutton Place near Guildford; the traditional country house became the centre of Getty Oil and his associated companies and he used the estate to entertain his British and Arabian friends (including the British Rothschild family and numerous rulers of Middle Eastern countries). Getty lived the rest of his life in the British Isles, dying of heart failure at the age of 83 on June 6, 1976.
Getty was married and divorced five times. He had five sons with four of his wives:
He was quoted as saying "A lasting relationship with a woman is only possible if you are a business failure".