## Can quantum transition state theory be defined as an exact *t* = 0 _{+} limit?

## Abstract

Here, the definition of the classical transition state theory (TST) as a *t* → 0 _{+} limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Contrary to this widely held view, Hele and Althorpe (HA) recently suggested that a true QTST can be defined as the exact *t* → 0 _{+} limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is themore »

- Authors:

- City Univ. of New York, Queens, NY (United States); City Univ. of New York, New York, NY (United States)
- Univ. of Chicago, Chicago, IL (United States)

- Publication Date:

- Research Org.:
- City Univ. of New York, New York, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1469687

- Alternate Identifier(s):
- OSTI ID: 1239449

- Grant/Contract Number:
- SC0001393

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 144; Journal Issue: 8; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Jang, Seogjoo, and Voth, Gregory A. Can quantum transition state theory be defined as an exact t = 0+ limit?. United States: N. p., 2016.
Web. doi:10.1063/1.4942482.
```

```
Jang, Seogjoo, & Voth, Gregory A. Can quantum transition state theory be defined as an exact t = 0+ limit?. United States. doi:10.1063/1.4942482.
```

```
Jang, Seogjoo, and Voth, Gregory A. Thu .
"Can quantum transition state theory be defined as an exact t = 0+ limit?". United States. doi:10.1063/1.4942482. https://www.osti.gov/servlets/purl/1469687.
```

```
@article{osti_1469687,
```

title = {Can quantum transition state theory be defined as an exact t = 0+ limit?},

author = {Jang, Seogjoo and Voth, Gregory A.},

abstractNote = {Here, the definition of the classical transition state theory (TST) as a t → 0+ limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Contrary to this widely held view, Hele and Althorpe (HA) recently suggested that a true QTST can be defined as the exact t → 0+ limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is the first important step in relating a t = 0+ limit to a physically measurable rate. Second, a theoretical analysis calls into question a key step in HA’s proof which appears not to rely on an exact quantum mechanical identity. The correction of this makes the true t = 0+ limit of HA’s QTST different from the RPMD-TST rate expression, but rather equal to the well-known path integral quantum transition state theory rate expression for the case of centroid dividing surface. An alternative quantum rate expression is then formulated starting from the linear response theory and by applying a recently developed formalism of real time dynamics of imaginary time path integrals. In conclusion, it is shown that the t → 0+ limit of the new rate expression vanishes in the exact quantum limit.},

doi = {10.1063/1.4942482},

journal = {Journal of Chemical Physics},

number = 8,

volume = 144,

place = {United States},

year = {2016},

month = {2}

}