Exhibit (a)(1)(E)

Offer to Purchase for Cash

All Outstanding Shares of Common Stock

of

Bronco Drilling Company, Inc.

at

$11.00 Per Share

Pursuant to the Offer to Purchase Dated April 26, 2011

by

Nomac Acquisition, Inc.

an indirect wholly owned subsidiary of

Chesapeake Energy Corporation

THE OFFER AND WITHDRAWAL RIGHTS EXPIRE AT 12:00 MIDNIGHT, NEW YORK CITY TIME, AT THE END OF MONDAY, MAY 23, 2011, UNLESS THE OFFER IS EXTENDED.

To Our Clients:

Enclosed for your consideration are the Offer to Purchase dated April 26, 2011 (the “Offer to Purchase”) and the related Letter of Transmittal (which, together with any amendments or supplements thereto, collectively constitute the “offer”) in connection with the tender offer by Nomac Acquisition, Inc., a Delaware corporation (“Purchaser”) and an indirect wholly owned subsidiary of Chesapeake Energy Corporation, an Oklahoma corporation (“Chesapeake”), to purchase for cash all outstanding shares of common stock, par value $0.01 per share, of Bronco Drilling Company, Inc. (the “Bronco shares”), a Delaware corporation (“Bronco”), at a purchase price of $11.00 per share in cash, without interest. Also enclosed is Bronco’s Solicitation/Recommendation Statement on Schedule 14D-9.

We are the holder of record of Bronco shares held for your account. A tender of such Bronco shares can be made only by us as the holder of record and pursuant to your instructions. The enclosed Letter of Transmittal is furnished to you for your information only and cannot be used by you to tender Bronco shares held by us for your account.

We request instructions as to whether you wish us to tender any or all of the Bronco shares held by us for your account, upon the terms and subject to the conditions set forth in the Offer to Purchase and the Letter of Transmittal.

Your attention is directed to the following:

1. The price paid in the offer is $11.00 per share in cash, without interest.

2. The offer is being made for all outstanding Bronco shares.

3. The offer is being made pursuant to an Agreement and Plan of Merger, dated as of April 14, 2011 (the “merger agreement”), by and among Chesapeake, Purchaser and Bronco. The merger agreement provides, among other things, that, on the terms and subject to the terms set forth in the merger agreement, after consummation of the offer, Purchaser will merge with and into Bronco (the “merger”), with Bronco continuing as the surviving corporation (the “surviving corporation”) and an indirect wholly owned subsidiary of Chesapeake. At the effective time of the merger (the “merger effective time”), each outstanding Bronco share (other than any Bronco shares held by Chesapeake, Purchaser, Bronco or any wholly owned subsidiary of Bronco

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or Chesapeake, and any Bronco shares held by stockholders who validly exercise their appraisal rights in connection with the merger) will be cancelled and extinguished and automatically converted into the right to receive the price per Bronco share paid in the offer, without interest.

4. The Board of Directors of Bronco has unanimously (i) declared that the merger agreement and the transactions contemplated by the merger agreement, including the offer and the merger, are fair to, advisable and in the best interests of Bronco and its stockholders and (ii) approved and declared advisable the merger agreement. THE BOARD OF DIRECTORS OF BRONCO RECOMMENDS THAT BRONCO STOCKHOLDERS ACCEPT THE OFFER AND TENDER THEIR BRONCO SHARES INTO THE OFFER.

5. The offer and withdrawal rights expire at 12:00 midnight, New York City time, at the end of Monday, May 23, 2011, unless the offer is extended by the Purchaser (as extended, the “Expiration Date”).

6. There is no financing condition to the offer. The offer is conditioned upon, among other things, (1) there being validly tendered in accordance with the terms of the offer, prior to the expiration date of the offer (as it may be extended from time to time) and not withdrawn, at least a majority of the total number of outstanding shares of Bronco common stock, on a fully diluted basis (which assumes conversion or exercise of all derivative securities of Bronco, regardless of the conversion or exercise price or other terms and conditions thereof) (excluding from the number of tendered Bronco shares, but not from the outstanding Bronco shares, Bronco shares tendered pursuant to the guaranteed delivery procedures that have not yet been delivered in settlement or satisfaction of such guarantee), (2) the expiration or termination of the applicable waiting period (and any extension thereof) under the Hart-Scott-Rodino Antitrust Improvements Act of 1976, as amended (the “Antitrust Condition”), and (3) there having not been imposed, in connection with satisfying the Antitrust Condition, any obligation of Chesapeake or any of its subsidiaries or affiliates to accept, commit to or effect the sale, divestiture or disposition of any assets or businesses of Chesapeake or Bronco or any of their respective subsidiaries or affiliates, or otherwise take or commit to take actions that after the closing of the transaction would limit Chesapeake’s freedom of action with respect to, or ability to retain, any portion of Chesapeake’s, Bronco’s or any of their subsidiaries’ or affiliates’ business or assets. A summary of the principal terms of the offer is described in the Offer to Purchase.

7. Any stock transfer taxes applicable to the sale of Bronco shares to Purchaser pursuant to the offer will be paid by Purchaser, except as otherwise set forth in Instruction 6 of the Letter of Transmittal. However, you may be subject to backup withholding at the applicable statutory rate, unless the required taxpayer identification information is provided and certain certification requirements are met, or unless an exemption is established. See Instruction 8 of the Letter of Transmittal.

If you wish to have us tender any or all of your Bronco shares, please complete, sign, detach and return to us the instruction form below. An envelope to return your instructions to us is enclosed. If you authorize tender of your Bronco shares, all such Bronco shares will be tendered unless otherwise specified on the instruction form. Your instructions should be forwarded to us in ample time to permit us to submit a tender on your behalf by the Expiration Date.

Payment for Bronco shares will be, in all cases, made only after such Bronco shares are accepted by Purchaser for payment pursuant to the offer and the timely receipt by Computershare Trust Company, N.A. (the “Depositary”) of (a) certificates for such Bronco shares or timely confirmation of a book-entry transfer of such Bronco shares into the Depositary’s account at the Book-Entry Transfer Facility (as defined in the Offer to Purchase) (a “Book-Entry Confirmation”) with respect to such Bronco shares, (b) a Letter of Transmittal, properly completed and duly executed, with any required signature guarantees (or, in the case of a book-entry transfer, an Agent’s Message (as defined in the Offer to Purchase) in lieu of the Letter of Transmittal), and (c) any other documents required by the Letter of Transmittal. Accordingly, tendering stockholders may be paid at different times, depending upon when certificates for Bronco shares or Book-Entry Confirmations with respect

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to Bronco shares are actually received by the Depositary. Under no circumstances will interest be paid on the consideration paid for Bronco shares pursuant to the offer, regardless of any extension of the offer or any delay in making payment for Bronco shares.

Purchaser is not aware of any jurisdiction where the making of the offer is prohibited by any administrative or judicial action pursuant to any valid state statute. If Purchaser becomes aware of any valid state statute prohibiting the making of the offer or the acceptance of the Bronco shares, Purchaser will make a good faith effort to comply with that state statute. If, after a good faith effort, Purchaser cannot comply with the state statute, Purchaser will not make the offer to, nor will Purchaser accept tenders from or on behalf of, the holders of Bronco shares in that state. In any jurisdiction where the securities, blue sky or other laws require the offer to be made by a licensed broker or dealer, the offer will be deemed to be made on behalf of Purchaser or by one or more registered brokers or dealers licensed under the laws of such jurisdiction.

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Instructions Form with Respect to

Offer to Purchase for Cash

All Outstanding Shares of Common Stock

of

Bronco Drilling Company, Inc.

$11.00 Per Share

Pursuant to the Offer to Purchase Dated April 26, 2011

by

Nomac Acquisition, Inc.

an indirect wholly owned subsidiary of

Chesapeake Energy Corporation

The undersigned acknowledge(s) receipt of your letter and the enclosed Offer to Purchase dated April 26, 2011 (the “Offer to Purchase”), and the related Letter of Transmittal, in connection with the offer by Nomac Acquisition, Inc., a Delaware corporation (“Purchaser”) and an indirect wholly owned subsidiary of Chesapeake Energy Corporation, an Oklahoma corporation, to purchase for cash all outstanding shares of common stock, par value $0.01 per share, of Bronco Drilling Company, Inc. (the “Bronco shares”), a Delaware corporation, at a purchase price of $11.00 per share in cash, without interest thereon, upon the terms and subject to the conditions set forth in the Offer to Purchase and the related Letter of Transmittal.

This will instruct you to tender the number of Bronco shares indicated below (or if no number is indicated below, all Bronco shares) held by you for the account of the undersigned, upon the terms and subject to the conditions set forth in the Offer to Purchase and in the related Letter of Transmittal furnished to the undersigned.

The undersigned understands and acknowledges that all questions as to validity, form, eligibility (including time of receipt) and acceptance of the surrender of any certificate representing Bronco shares submitted on my behalf to Computershare Trust Company, N.A. (the “Depositary”) will be determined by Purchaser in its sole and absolute discretion (provided that Purchaser may delegate such power in whole or in part to the Depositary).

Number of Bronco shares to be Tendered:

shares*	SIGN HERE
Dated                     , 20
Signature(s)
Name(s)
Address(es)
(Zip Code)
Area Code and Telephone Number
Taxpayer Identification or Social Security No.

* Unless otherwise indicated, it will be assumed that all Shares held for the undersigned’s account are to be tendered.

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