Similarity transformations

Let us note that for the Schrodinger equation Hiff =E4f,we can perform an interesting sequence of transformations based on the wave operator e  

We obtain the eigenvalue equation again, but for the similariiy transformed Hamiltonian 

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The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

It also follows that A = A, which is a characteristic of similarity transformations, so A HA = E implies that... 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

This identity illustrates a so-ealled similarity transform of M using S. Sinee A is diagonal, we say that the similarity transform S diagonalizes M. Notiee this would still work if we... 

Based on the above similarity transform, we ean now show that the traee of a matrix (i.e., the sum of its diagonal elements) is independent of the representation in whieh the matrix is formed, and, in partieular, the traee is equal to the sum of the eigenvalues of the matrix. The proof of this theorem proeeeds as follows ... 

We will see later that performing a similarity transform expresses a matrix in a... 

The first report on the Pd(II)-promoted Cope rearrangement is the conversion of fw./ra/w-l,5-cyclodecadiene (44) into c/5-l,2-divinylcyclohexane-PdCl2 complex (45) with a stoichiometric amount of PdCl2(PhCN)2 at room temperature. The complex formation is the driving force of this unusual rearrangement [38,39]. A similar transformation of germacrane (l,5-dimethyl-8-isopropyli-dene-/rfflu,/ra j-l,5-cyclodecadiene) takes place[40j. 

The high cost of SF and the incomplete use of fluorine justify its use only for inaccessible ben2otrifluorides. The related Hquid S—F reagent, (diethylarnino)sulfur trifluoride (DAST), (C2H )2NSF2, also effects similar transformations with aromatic carboxyhc acids (108). 

Except for helium, all of the elements in Group 18 free2e into a face-centered cubic (fee) crystal stmeture at normal pressure. Both helium isotopes assume this stmeture only at high pressures. The formation of a high pressure phase of soHd xenon having electrical conductivity comparable to a metal has been reported at 33 GPa (330 kbar) and 32 K, and similar transformations by a band-overlap process have been predicted at 15 GPa (150 kbar) for radon and at 60 GPa (600 kbar) for krypton (51). 

Moreover, fermentation of various a-substituted cycloalkanone enol esters results in optically active six-, eight-, ten-, and twelve-membered ring ketones with 70—96% ee (84). Isolated enzymes catalyze similar transformations, bacillus coagulans and Candida glindracea]i 2Lse OF (Meito Sangyo) hydrolyze a number of cycHc and acycHc enol esters, giving ketones in 40—80% yield and 14—85% ee (85,86). 

Similar transformations are based upon 1,2-thiazine 2,2-dioxides (5-sultones), easily obtained by treating a,/3- or /3,y-unsaturated ketones with acetic anhydride and sulfuric acid. These compounds can be converted subsequently into furans, thiophenes or pyrroles (Scheme 115). 

Earlier reported syntheses have been shown to give isoxazolin-5-ones. Other isoxazolin-3-ones have been prepared by the reaction of methylacetoacetic esters and hydroxylamine. An additional synthesis was reported by the action at 0°C of hydroxylamine on ethyl -benzoylpropionate to produce an insoluble hydroxamic acid which cyclized on acid treatment. The hydroxamic acid acetal was similarly transformed into the isoxazolin-3-one (Scheme 149) (71BSF3664, 70BSF1978). 

Group Method The type of transformation can be deduced using group theory. For a complete exposition, see Refs. 9, 12, and 145 a shortened version is in Ref. 106. Basically, a similarity transformation should be considered when one of the independent variables has no physical scale (perhaps it goes to infinity). The boundary conditions must also simphfy (and combine) since each transformation leads to a differential equation with one fewer independent variable. 

Note that the length dimension goes to infinity, so that there is no length scale in the problem statement this is a clue to try a similarity transformation. The transformation examined here is... 

The thiation procedure described here is an example of a general synthetic method for the conversion of carbonyl to thiocarbonyl groups. Similar transformations have been carried out with ketones, carboxamides,esters,thioesters, 1 actones, " thiol actones, - imides, enaminones, and protected peptides. ... 

A similar transformation results when trimethylsilyloxy-substituted allylic halides react with silver perchlorate in nitromethane. The resulting allylic cation gives cycloaddition reactions with dienes such as cyclopentadiene. The isolated products result from desilyla-tion of the initial adducts ... 

Bartonand Wolft achieved a similar transformations after introduction of a 5,6-double bond by zinc reduction of a 5,6-halohydrin. 

A similar transformation to a metallic, tetragonal -form can be effected in Si and Ge by subjecting them to pressures of 200 and I20kbar respectively along the c-axis, and again the density increases by 25% from the value at atmospheric pressure. Lead is familiar as a blue-grey, malleable metal with a fairly high density (nearly 5 times that of Si and twice those of Ge and Sn, but only half that of Os and Ir). 

One can also use diethyl phosphorocyanidate (DEPC) to effect a similar transformation. This was the reagent of choice in the generation of l-substituted-4-oxygenated-P-carbolines (49— 50). 

The sequence (81 84) has been proposed - to account for this process which involves decomposition of the aci-nitro anion by strong acid. The a-carbonyl group presumably stabilizes the aci-salt and thus could be responsible for inhibiting the normal Nef reaction. A similar transformation has been observed in the case of a 16-nitro-17-0X0 steroid. 

Catalytic dehydrogenation using palladium/maleic acid has been used to convert tetrahydro-j8-earbolines of general structure 142 into the dihydro-jS Carbolinium salts (143) A similar transformation has been carried out by oxidation with iodine. ... 

Popov et al. (75KGS1678) have described similar transformations for /3-(l-methyl-3-indazolyl)-Q -bromoacrylic and /3-(2-methyl-3-indazolyl)-Q -bromo-acrylic acids (Table TX). The treatment of the bromoacrylic acids with alcohol... 

Similar transformations were also found on heating of 2-(but-3-ynylthio)pyrimidine (X = S) and 2-(but-3-ynylsulfonyl)pyrimidine (X = S02), 2,3-dihydrothieno[2,3-h]pyridine and l,l-dioxo-2,3-dihy-drothieno[2,3-h]pyridine, respectively being obtained (Scheme 35). 

Thus, acetylation of aniline affords acetanilide (20), an analgesic widely used in proprietary headache remedies. A similar transformation on p-aminophenol gives the analgesic, acetaminophen (21). It is of interest that the latter is also formed in vivo on administration of 21. An interesting preparation of this drug involves Schmidt rearrangement of the hydrazone (24) from p-liydroxyacetophenone. ... 

Similar transformations have not as yet been successfully applied to the tetracyclines bearing a hydroxy group at Cs, and no mutant culture has been reported that biosynthesizes a 6-deoxy-5-oxytetracycline. However, other means have been found to avoid 5a,6-dehydration in this subfamily. Treatment of 3 with N-... 

Since the topology of Gf, is invariant under similarity transformations of L, I its complete structure can be computed using L, It remains only to find the cycle sum of a general block C[ ef) ]. We quote the result ([biggs74], [evct79]) ... 

Subsequent to the announcement (47) of this reaction, a similar transformation of 4,6-O-benzylidene acetals was reported (35) in which a free radical initiator was used. Under these conditions a blocked derivative such as 93 (R = Bz) was transformed in 76% yield into the corresponding 94. Selective formation of the 6-bromo derivative was observed as in the initial studies (47). 

Mioskowski et al. examined the reductive alkylation of simple epoxides by or-ganolithiums in THF in considerable detail, and found that the best yields and stereoselectivities were obtained with secondary and tertiary alkyllithiums (Table 5.2, Entries 1-5) [42]. n-BuLi gave a mixture of olefins (Entry 6), whereas PhLi and MeLi (Entries 7 and 8) gave very poor yields. Similar transformations have been reported with the use of lithium tetraalkylcerate reagents (Entries 9 and 10) [43]. 

A similar transformation of (l-methoxy-2-propenyl)diphenylphosphane oxide leads to (fyS-mcthoxy-l, 3-dienes61. 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

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